A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs without 4-cycles with maximum degree \(\Delta\geq10\).
All graphs considered are finite, simple and undirected. Let \(G\) be a graph. We use \(V(G)\), \(E(G)\), \(\Delta(G)\) and \(\delta(G)\) to denote its vertex set, edge set, maximum degree and minimum degree, respectively. For a vertex \(v\in V(G)\), \(N_{G}(v)\) denotes the set of vertices that are adjacent to \(v\) in \(G\). By \(d(v):=|N_{G}(v)|\) denotes the \(degree\) of \(v\) in \(G\). For planar graphs \(G\), \(F(G)\) denotes its face set, the degree of a face \(f\), denoted by \(d(f)\), is the length of a boundary walk around \(f\) in \(G\). We call \(v\) a \(k\)-vertex, or a \(k^{+}\)-vertex, or a \(k^{-}\)-vertex if \(d(v)=k\), or \(d(v)\geq k\), or \(d(v)\leq k\) respectively and call \(f\) a \(k\)-face, or a \(k^{+}\)-face, or a \(k^{-}\)-face if \(d(f)=k\), or \(d(f)\geq k\), or \(d(f)\leq k\) respectively. Any undefined notation follows that of Bondy and Murty [2]. A \(total-k-coloring\) of a graph \(G\) is a coloring of \(V(G)\cup E(G)\) using \(k\) colors such that no two adjacent or incident elements receive the same color. The \(total\,chromatic\) \(number\) \(\chi''(G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has a total-\(k\)-coloring. It is clearly that \(\chi''(G)\geq\Delta(G) + 1\). Behzad and Vizing [1,6] posed independently the conjecture, \(\chi''(G)\leq\Delta(G) + 2\) for any graph \(G\), which is known as the total coloring conjecture.
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. The notion of a 1-planar graph was introduced by Ringel [4] in connection with the problem of simultaneous coloring of adjacent/incident vertices and faces of plane graphs. In [10], Zhang et al. proved that every 1-planar graph with maximum degree \(\Delta(G)\geq16\) is totally \((\Delta(G)+2)\)-choosable, which implies that the total-coloring conjecture holds for 1-planar graphs with maximum degree at least 16. Later, Czap [3] proved (Without discharging method) that for every 1-planar graph \(G\) with \(\Delta(G)\geq10\) it holds \(\chi''(G)\leq\Delta(G) + 3\). Moreover, if \(\chi(G)\leq4\), then \(\chi''(G)\leq\Delta(G) + 2\). In the same paper, the author also verified that for every 1-planar graph \(G\) without adjacent triangles and with \(\Delta(G)\geq10\) it holds \(\chi''(G)\leq\Delta(G) + 3\). Moreover, if \(\chi(G)\leq4\), then \(\chi''(G)\leq\Delta(G) + 2\). Zhang and Hou [7] showed the following theorem which improve the lower bound for the maximum degree in the corollary of [10] to 13.
Recently, Sun and Wu [5] verified the total coloring \(\chi''(G)\leq r + 2\), for every 1-planar graph \(G\) if \(\Delta(G)\geq9\) and \(g(G)\geq4\) where \(\Delta(G)\) is the maximum degree of \(G\) and \(g(G)\) is the girth of \(G\)
Theorem 1.1. Let \(G\) be a 1-planar graph with maximum degree \(\Delta(G)\) and let \(r\) be an integer. If \(\Delta\leq r\) and \(r \geq13\), then \(\chi''(G)\leq r + 2\).
In this paper, we shall prove the following results:
Theorem 1.2. Let \(G\) be a 1-planar graph without 4-cycles, with maximum degree \(\Delta(G)\geq10\). Then \(\chi''(G)\leq \Delta(G) + 2\).
Let \(G\) in this paper has been embedded on a plane such that every edge is crossed by at most one other edge and the number of crossings is as small as possible. The \(associated\) \(plane\) \(graph\) \(G^{\times}\) of \(G\) is obtained by turning all crossings of \(G\) into new 4-vertices on a plane. For convenience, a vertex in \(G^{\times}\) is called \(false\) if it is not a vertex of \(G\) and \(real\) otherwise. A \(false\) \(face\) means a face \(f\) in \(G^{\times}\) that is incident with at least one \(false\) \(vertex\); otherwise, \(f\) is a \(normal\) \(face\). For a vertex \(v\in V(G^{\times})\), we use \(f_{k}(v)\) to denote the number of \(k\)-faces which are incident with \(v\), \(n_{i}(v)\) to denote the number of \(i\)-vertices which are adjacent to \(v\), and \(n_{c}(v)\) to denote the number of false vertices which are adjacent to \(v\).
For convenience, we use \(v_{1}, v_{2}, \cdot\cdot\cdot, v_{d}\) to denote the neighbors of a \(d\)-vertex \(v\) in \(G^{\times}\) that occur around it in a clockwise order. By \(f_{i}\) denote the face incident with \(vv_{i}\) and \(vv_{i+1}\) in \(G^{\times}\), where the addition on subscripts are taken modulo \(d\).
Let \(G\) be a counterexample with \(|E(G)|\) as small as possible to Theorem 1.2. By minimality of \(G\) we can assume that it is connected and that it has no total (\(\Delta(G)\) + 2)-colorings. First we investigate some structural of properties of \(G\). Here, we give some known lemmas.
Lemma 2.1. .[10] Let \(uv\) be an edge in \(G\). If \(\min \{d_{G}(u), d_{G}(v)\} \leq \lfloor\frac{\Delta+1}{2}\rfloor\), then \(d_{G}(u) + d_{G}(v) \geq\Delta + 3\).
From this lemma, we deduce that \(\delta(G)\geq3\).
Lemma 2.2. [7] Let \(V_{i}\) be the set of \(i\)-vertices in \(G\). We have \(|V_{\Delta}| > 2|V_{3}|\).
Lemma 2.3. Let \(G\) be a 1-planar graph without 4-cycles and let \(G^{\times}\) be its associated plane graph. Then for every \(5^{+}\) vertex \(v\in V(G)\), \(v\) is incident with at most \(\lfloor\frac{4}{5}d_{G}(v)\rfloor\) 3-faces in \(G^{\times}\).
The proof is just similar to the one in [8], with only quite a little minor changes. So we omit it here and refer the reader to Lemma 4 of [8].
Lemma 2.4. [9] Let \(G\) be a 1-plane graph and let \(G^{\times}\) be its associated plane graph.Then the following hold:
1) For any two false vertices \(u\) and \(v\) in \(G^{\times}\), \(u\) and \(v\) are not adjacent in \(G^{\times}\).
2) If \(d_{G}(u)=3\) and \(v\) is a false vertex in \(G^{\times}\), then either \(u\) and \(v\) are not adjacent in \(G^{\times}\), or \(uv\) is not incident with two 3-faces.
3) Let \(v\) be a 3-vertex in \(G\). If \(v\) is incident with two false 3-faces \(vv_{1}v_{2}\) and \(vv_{1}v_{3}\) in \(G^{\times}\), then \(v_{2}\) and \(v_{3}\) are both false and \(v\) is incident with a \(5^{+}\)-face in \(G^{\times}\).
Lemma 2.5. Let \(G\) be a 1-plane graph and let \(G^{\times}\) be its associated plane graph. Then, every 5-face in \(G^{\times}\) is incident with at most four \(5^{-}\)-vertices.
The proof is just similar to the one in [7], with only quite a little minor changes. So we omit it here and refer the reader to Lemma 9 of [7].
Lemma 2.6. [10] For each integer \(3 \leq k\leq 5\), let \(X_{k} = \{x \in V (G) | d_{G}(x)\leq k\}\), \(Y_{k}= \bigcup_{x\in X_{k}}N_{G}(x)\). If \(X_{k}\neq \emptyset\), then there exists a bipartite subgraph \(M_{k} = (X_{k},Y_{k})\) of \(G\) such that \(d_{M_{k}}(x) = 1\) for any \(x\in X_{k}\) and \(d_{M_{k}}(y) \leq k-2\) for any \(y \in Y_{k}\). We call \(y\) the \(k\)-master of \(x\) if \(xy\in M_{k}\) and \(x\in X_{k}\).
By this lemma, we deduce that each \(k\)-vertex (\(3 \leq k\leq 5\)) has a \(j\)-master (\(k \leq j\leq 5\)).
Lemma 2.7. [7] Let \(G\) be a 1-plane graph and let \(v\) be a vertex of \(G\). If \(d_{G}(v)=3\), then, \(v\) cannot be contained in a triangle in \(G\). If \(d_{G}(v)=4\) with \(N_{G}(v)=v_{1}, v_{2},v_{3}, v_{4}\), then, for any \(i\),(\(1 \leq i\leq 4\)), the edge \(vv_{i}\) can not be contained in two triangles.
Lemma 2.8. Let \(G\) be a 1-plane graph without 4-cycles and \(G^{\times}\) be its associated plane graph. Let \(v\) be a vertex of \(G\), then, there are no five consecutive 3-faces that are incident with \(v\) in \(G^{\times}\). If \(v\) is incident with \(i\) consecutive 3-faces \(f_{1}, f_{2}, \cdot\cdot\cdot, f_{i}\), (\(3 \leq i\leq 4\)) in \(G^{\times}\), then, there is at most a real small vertex among the neighbors of \(v\) on these consecutive 3-faces. Moreover, if \(v\) is incident with 4 consecutive 3-faces \(f_{1}, f_{2}, f_{3}, f_{4}\), then \(v_{1}, v_{3}, v_{5}\) are false vertices, \(v_{2}, v_{4}\) are real vertices.
The proof is just similar to the one in [8], with only quite a little minor changes. So we omit it here and refer the reader to Lemma 4 of [8].
Then, we begin to prove the main result of the paper.
A vertex \(v\) in \(G\) is \(small\) if \(d(v)\leq 5\) and is \(big\) if \(d(v)\geq6\). Note that the degree of a false vertex in \(G^{\times}\) is four, so every false vertex is small.
In the following, we apply the discharging method on associated 1-planar graph \(G^{\times}\) of \(G\) and complete the proof by a contradiction. Since \(G^{\times}\) is a plane graph, we have \[\sum\limits_{v \in V\left( {{G^ \times }} \right)} {\left( {d\left( v \right) – 6} \right) + \sum\limits_{f \in F\left( {{G^ \times }} \right)} {\left( {2d\left( f \right) – 6} \right) = – 12} },\] by the well-known Euler’s formular. Now we define the initial charge function \({ch(x)}\) of \(x \in V\left( {{G^ \times }} \right) \cup F\left( {{G^ \times }} \right)\). Let \(ch(v) = d(v) – 6\ \) if \(x\in V\left( {{G^ \times }} \right)\) and \(ch(f) = 2d(f) – 6\) if \(x \in F\left( {{G^ \times }} \right)\). And we define suitable discharging rules below to change the initial charge function \({ch(x)}\) to the final charge function \(ch'(x)\) on \(V\left( {{G^ \times }} \right) \cup F\left( {{G^ \times }} \right)\). Then we still have \(\sum\nolimits_{x \in V({G^ \times }) \cup F({G^ \times })} {ch'(x)} = \sum\nolimits_{x \in V({G^ \times }) \cup F({G^ \times })} {ch(x)} = – 12\), since any discharging procedure preserves the total charge of \(G^{\times}\).
Our discharging rules are defined as follows.
R1. Each \(f\) in \(G^{\times}\) where \(d(f)\geq4\) sends \(\frac{2d(f)-6}{t(f)}\) to each small vertex incident with it, where \(t(f)\) is the number of small vertices incident with the face \(f\).
R2. Each 3-vertex in \(G\) receives \(\frac{2}{9}\) from its \(i\)-master (\(3 \leq i\leq 5\)).
R3. Each 4-vertex in \(G\) receives \(\frac{6}{25}\) from its \(i\)-master (\(4 \leq i\leq 5\)).
R4. Each \(\Delta\)-vertex gives \(\frac{1}{2}\) to a common pot from which each 3-vertex receives 1, if \(|V_{3}| > 0\).
R5. Let \(w\) be a false vertex and \(w\) is incident with a 3-face \(f\) in \(G^{\times}\), then each \(8^{+}\)-neighbor of \(w\) on \(f\) sends \(\frac{13}{50}\) to \(w\).
R6. Let \(w\) be a real 4-vertex and \(w\) is incident with a normal 3-face \(f\) in \(G^{\times}\), then each \(8^{+}\)-neighbor of \(w\) on \(f\) sends \(\frac{13}{50}\) to \(w\).
R7. Let \(u\) be a real \(4\) -vertex, \(v\) is a false vertex in \(G^{\times}\), \(uv\in E(G^{\times})\) and \(uv\) is incident with two 3-faces in \(G^{\times}\), then \(v\) sends \(\frac{13}{25}\) to \(u\).
R8. If a false vertex \(v\) in \(G^{\times}\) is incident with four \(4^{+}\)-faces in \(G^{\times}\), then \(v\) sends \(\frac{5}{12}\) to each 4-neighbor of \(v\).
R9. If a false vertex \(v\) in \(G^{\times}\) is incident with exactly one 3-face \(f\) in \(G^{\times}\), then \(v\) sends \(\frac{1}{3}\) to its 3-neighbor on \(f\).
R10. Let \(v\) be a 3-vertex and \(v\) is not incident with any 3-face in \(G^{\times}\), then \(v\) sends \(\frac{1}{6}\) to each false vertex which is adjacent to \(v\).
R11. If a real 4-vertex \(v\) in \(G^{\times}\) is incident with four \(4^{+}\)-faces in \(G^{\times}\), then \(v\) sends \(\frac{11}{75}\) to each false vertex which is adjacent to \(v\).
R12. If a false vertex \(u\) in \(G^{\times}\) is adjacent to a 5-vertex \(v\) in \(G^{\times}\), and \(uv\) is incident with \(4^{+}\)-faces \(f_{1}\) and \(f_{2}\) which are adjacent in \(G^{\times}\), then \(v\) sends \(\frac{1}{6}\) to \(u\).
In the following, we check that the final charge \(ch'(x)\) on each vertex and face is nonnegative, and we also show the final charge of the common pot is nonnegative. This implies that \(\sum\nolimits_{x \in V({G^ \times }) \cup F({G^ \times })} {ch'(x)}\geq0\) for all \(x \in V\left( {{G^ \times }} \right) \cup F\left( {{G^ \times }} \right)\), a contradiction. This completes the proof of Theorem 1.2.
First of all, by R4, the final charge of the common pot is at least \(\frac{1}{2}|V_{\Delta}|-|V_{3}| > 0\) since \(|V_{\Delta}| >2|V_{3}|\) by Lemma 2.2. One can also check that the final charge of every face in \(F(G^{\times})\) is nonnegative by R1. Thus in the following we consider the vertices in \(G^{\times}\).
Case 1. \(d=3\). By R2 and R4, \(v\) receives 1 from the common pot and \(\frac{2}{9}\times3=\frac{2}{3}\) from its \(i\)-masters, where \(3\leq i\leq5\). Since \(G\) is a 1-planar graph without 4-cycles, \(v\) is incident with at most two 3-faces in \(G^{\times}\) by Lemma 2.4 and Lemma 2.7. Now, we consider three subcases.
Case 1.1. If \(v\) is not incident with any 3-face in \(G^{\times}\), then \(f_{1}, f_{2}, f_{3}\) are all \(4^{+}\)-faces.
First, assume that \(v\) is incident with at least one \(5^{+}\)-face, without loss of generality, assume that \(f_{1}\), then \(v\) would receive at least 1 from \(f_{1}\), and \(\frac{2}{4}\times2=1\) from \(f_{2},f_{3}\), by Lemma 2.5 and R1. By R10, \(v\) sends at most \(\frac{1}{6}\times3=\frac{1}{2}\) to false vertices which are adjacent to \(v\). Thus, \(ch'(v)\geq-3+1+\frac{2}{3}+1+1-\frac{1}{2}>0\).
Second, assume that \(f_{1}, f_{2}, f_{3}\) are all 4-faces. If \(v\) is adjacent to at least one real vertex in \(G^{\times}\), say \(v_{1}\), then \(d(v_{1})\geq10\), thus \(f_{1}, f_{3}\) sends at least \(\frac{2}{3}\times2=\frac{4}{3}\) to \(v\), and \(f_{2}\) sends at least \(\frac{2}{4}=\frac{1}{2}\) to \(v\) by R1. By R10, \(v\) sends at most \(\frac{1}{6}\times3=\frac{1}{2}\) to false vertices which are adjacent to \(v\). Thus,\(ch'(v)\geq-3+1+\frac{2}{3}+\frac{4}{3}+\frac{1}{2}-\frac{1}{2}=0\). Otherwise, \(v_{1}, v_{2}, v_{3}\) are all false vertices. Let \(x_{i}\) be the fourth(undefined) vertices of the 4-faces \(f_{i}\) \((i=1,2,3)\). It is easy to check that \(x_{1}x_{2},x_{2}x_{3},x_{3}x_{1}\)\(\in\)\(E(G)\) by the drawing of \(G\). Since \(f_{1}, f_{2}, f_{3}\) are all 4-faces, there are at least two big vertices among \(x_{1}, x_{2}, x_{3}\) by Lemma 2.1, without loss of generality, assume that \(x_{1},x_{2}\), thus, \(f_{1}, f_{2}\) send at least \(\frac{2}{3}\times2=\frac{4}{3}\) to \(v\), and \(f_{3}\) sends at least \(\frac{2}{4}=\frac{1}{2}\) to \(v\) by R1. By R10, \(v\) sends at most \(\frac{1}{6}\times3=\frac{1}{2}\) to false vertices which are adjacent to \(v\). Thus, \(ch'(v)\geq-3+1+\frac{2}{3}+\frac{4}{3}+\frac{1}{2}-\frac{1}{2}=0\).
Case 1.2. If \(v\) is incident with exactly one 3-face in \(G^{\times}\), then without loss of generality assume that \(f_{3}\) is a 3-face. Since no two false vertices are adjacent in \(G^{\times}\) by Lemma 2.4, there is a real vertex, among \(v_{1}\) and \(v_{3}\), say \(v_{1}\), then \(d(v_{1})\geq10\).
Assume that \(v_{2}\) is also a real vertex, then \(d(v_{2})\geq10\). Thus, \(f_{1}\) sends at least 1 to \(v\), \(f_{2}\) sends at least \(\frac{2}{4-1}=\frac{2}{3}\) to \(v\) by R1, Thus,\(ch'(v)\geq-3+1+\frac{2}{3}+1+\frac{2}{3}>0\). Otherwise, \(v_{2}\) is a false vertex. Let \(x_{i}\) be the second neighbors of \(v_{2}\) on \(f_{i}\) \((i=1,2)\), it is easy to check that \(x_{1}x_{2}\)\(\in\)\(E(G)\) by the drawing of \(G\). Thus, at least one of \(x_{1}\) and \(x_{2}\) is big by Lemma 2.1. This implies that \(v\) receives at least \(\min\{1+\frac{1}{2}, \frac{2}{3}\times2\}\)\(=\frac{4}{3}\), from \(f_{1}\) and \(f_{2}\) by R1. Therefore, \(ch'(v)\geq-3+1+\frac{2}{3}+\frac{4}{3}=0\).
Case 1.3. If \(v\) is incident with exactly two 3-faces in \(G^{\times}\), then without loss of generality assume that \(f_{2}\) and \(f_{3}\) are 3-faces. By Lemma 2.4 and Lemma 2.7, \(v_{3}\) must be a real vertex, \(v_{1}\) and \(v_{2}\) are false vertices, and \(f_{1}\) is a \(5^{+}\)-face. Thus, \(f_{1}\) sends at least \(\frac{4}{5-1}=1\) to \(v\) by Lemma 2.5 and R1. Since \(G\) is a 1-planar graph without 4-cycles, so, there is at least a vertex among \(v_{1}\) and \(v_{2}\) which is incident with exactly one 3-face, say \(v_{2}\). Then, \(v_{2}\) sends \(\frac{1}{3}\) to \(v\) by R9. Thus, \(ch'(v)\geq-3+1+\frac{2}{3}+1+\frac{1}{3}=0\).
Case 2. \(d=4\) and \(v\) is a real vertex, then \(v\) has one 4-master and one 5-master. So \(v\) receives totally \(\frac{6}{25}\times2=\frac{12}{25}\) from its masters by R3. Since \(G\) is a 1-planar graph without 4-cycles, \(v\) is incident with at most three 3-faces in \(G^{\times}\) by Lemma 2.7.
If \(v\) is incident with exactly one 3-face in \(G^{\times}\), say \(f_{1}\), then there is at most one false vertex among \(v_{1}\) and \(v_{2}\) by Lemma 2.4. Suppose that \(v_{1}\) is a false vertex, then, \(d(v_{2})\geq9\) by Lemma 2.1, thus, \(v\) receives at least \(\frac{2}{4-1}=\frac{2}{3}\) from \(f_{2}\), receives \(\frac{1}{2}\times2=1\) from \(f_{3}\) and \(f_{4}\) by R1. Therefore, \(ch'(v)\geq-2+\frac{12}{25}+\frac{2}{3}+1=\frac{11}{75}\).
If \(v\) is not incident with any 3-face, then \(v\) is incident with four \(4^{+}\)-faces.
First, assume that \(v\) is incident with at least one \(5^{+}\)-face, say \(f_{1}\), then, \(v\) receives at least 1 from \(f_{1}\) by Lemma 2.5 and R1, receives \(\frac{1}{2}\times3=\frac{3}{2}\) from \(f_{2}\), \(f_{3}\) and \(f_{4}\) by R1. \(v\) sends at most \(\frac{11}{75}\times4=\frac{44}{75}\) to false vertices that are adjacent to \(v\) by R11. Therefore, \(ch'(v)\geq-2+\frac{12}{25}+1+\frac{3}{2}-\frac{44}{75}>0\).
Second, assume that \(v\) is incident with four 4-faces, if the neighbors of \(v\) are all false vertices, then, let \(x_{i}\) be the fourth(undefined) vertices of the 4-faces \(f_{i}\) \((i=1,2,3,4)\). It is easy to check that \(x_{1}x_{2},x_{3}x_{4}\)\(\in\)\(E(G)\) by the drawing of \(G\). Thus, at least one of \(x_{1}\) and \(x_{2}\) is big, similarly to \(x_{3}\) and \(x_{4}\) by Lemma 2.1. This implies that \(v\) receives at least \(\frac{1}{2}\times2+\frac{2}{3}\times2=\frac{7}{3}\) from \(f_{1}\), \(f_{2}\),\(f_{3}\) and \(f_{4}\) by R1. \(v\) sends at most \(\frac{11}{75}\times4=\frac{44}{75}\) to false vertices that are adjacent to \(v\) by R11. Therefore, \(ch'(v)\geq-2+\frac{12}{25}+\frac{7}{3}-\frac{44}{75}=\frac{17}{75}\). Otherwise, \(v\) is adjacent to at least one real vertex. Thus, \(v\) receives at least \(\frac{1}{2}\times4=2\) from \(f_{1}\), \(f_{2}\),\(f_{3}\) and \(f_{4}\) by R1. \(v\) sends at most \(\frac{11}{75}\times3=\frac{33}{75}\) to false vertices that are adjacent to \(v\) by R11. Therefore, \(ch'(v)\geq-2+\frac{12}{25}+2-\frac{33}{75}=\frac{3}{75}\).
If \(v\) is incident with exactly three 3-faces, then without loss of generality assume that \(f_{1}\) \(f_{2}\) and \(f_{4}\) are 3-faces. Since \(G\) is a 1-planar graph without 4-cycles, so, \(v\) is adjacent to exactly two false vertices by Lemma 2.4 and Lemma 2.7 in \(G^{\times}\), and moreover \(f_{3}\) is a \(5^{+}\)-face. First, assume that two false vertices are not adjacent, say \(v_{1}\) and \(v_{3}\), then \(d(v_{2})\geq9\), \(d(v_{4})\geq9\) by Lemma 2.1. Then, \(f_{3}\) sends at least 1 to \(v\) by R1, \(v_{1}\) sends \(\frac{13}{25}\) to \(v\) by R7. Thus, \(ch'(v)\geq-2+\frac{12}{25}+1+\frac{13}{25}=0\).
Second, assume that two false vertices are \(v_{3}\) and \(v_{4}\), then, \(d(v_{1})\geq9\), \(d(v_{2})\geq9\) by Lemma 2.1. Thus, \(f_{3}\) sends at least 1 to \(v\) by R1, \(v_{1}\) and \(v_{2}\) send \(\frac{13}{50}\times2=\frac{13}{25}\) to \(v\) by R6. This implies that \(ch'(v)\geq-2+\frac{12}{25}+1+\frac{13}{25}=0\).
If \(v\) is incident with exactly two 3-faces in \(G^{\times}\), we consider four subcases.
Case 2.1. If \(v\) is not adjacent to any false vertex, then \(v_{i}\geq9\) \((i=1,2,3,4)\) by Lemma 2.1, and the two 3-faces that are incident with \(v\) have no common edge by Lemma 7, without loss of generality assume that \(f_{2}\) and \(f_{4}\) are 3-faces. Then, \(v\) receives a total of \(\frac{2}{4-2}\times2=2\) from \(f_{1}\) and \(f_{3}\), thus, \(ch'(v)\geq-2+\frac{12}{25}+2>0\).
Case 2.2. If \(v\) is adjacent to exactly one false vertex, without loss of generality assume that \(v_{1}\), then \(d(v_{i})\geq9\) \((i=2,3,4)\) by Lemma 2.1. First, assume that the two 3-faces that are incident with \(v\) have no common edge, say \(f_{2}\) and \(f_{4}\), then, \(v\) receives at least \(\frac{2}{4-2}=1\) from \(f_{3}\), receives at least \(\frac{2}{4-1}=\frac{2}{3}\) from \(f_{1}\) by R1, and \(v\) receives \(\frac{13}{50}\times2\) from \(v_{2}\) and \(v_{3}\) by R6. Thus, \(ch'(v)\geq-2+\frac{12}{25}+1+\frac{2}{3}+\frac{13}{50}\times2=\frac{2}{3}\). Second, assume that the two 3-faces that are incident with \(v\) have one common edge, since \(G\) has no 4-cycles, then, \(v_{1}\) is incident with at least one 3-face. If \(v_{1}\) is incident with exactly one 3-face, without loss of generality assume that \(f_{1}\), then \(f_{2}\) is a real 3-face in \(G^{\times}\). By R6, \(v\) receives \(\frac{13}{50}\times2\) from \(v_{2}\) and \(v_{3}\), \(v\) receives at least \(\frac{2}{4-2}=1\) from \(f_{3}\) and receives at least \(\frac{2}{4-1}=\frac{2}{3}\) from \(f_{4}\) by R1. Thus, \(ch'(v)\geq-2+\frac{12}{25}+1+\frac{2}{3}+\frac{13}{50}\times2=\frac{2}{3}\). If \(v_{1}\) is incident with two 3-faces, say \(f_{1}\) and \(f_{4}\), then, \(v\) receives at least \(\frac{2}{4-2}\times2=2\) from \(f_{2}\) and \(f_{3}\). Therefore, \(ch'(v)\geq-2+\frac{12}{25}+2=\frac{12}{25}\).
Case 2.3. If \(v\) is adjacent to exactly two false vertices.
First, assume that two faces which are incident with \(v\) are not adjacent, say \(f_{2}\) and \(f_{4}\) are both 3-faces, then, \(f_{1}\) and \(f_{3}\) are both \(4^{+}\)-faces. If two false vertices that are adjacent to \(v\) are incident with the same \(4^{+}\)-face, say \(f_{1}\), then, \(v_{1}\) and \(v_{2}\) are both false vertices, \(v_{3}\) and \(v_{4}\) are both big vertices. Since \(G\) has no 4-cycles, then, \(f_{1}\) is a \(5^{+}\)-face. It implies that \(f_{1}\) sends at least 1 to \(v\) and \(f_{3}\) sends at least \(\frac{2}{4-2}=1\) to \(v\) by Lemma 2.5 and R1. Thus, \(ch'(v)\geq-2+\frac{12}{25}+1+1=\frac{12}{25}\). Otherwise, two false vertices that are adjacent to \(v\) are incident with different \(4^{+}\)-faces, say \(f_{1}\) and \(f_{3}\), since \(G\) has no 4-cycles, then, \(f_{1}\) and \(f_{3}\) are \(5^{+}\)-faces. Thus, \(v\) receives at least \(\frac{4}{5-1}\times2=2\) from \(f_{1}\) and \(f_{3}\) by Lemma 2.5 and R1. Therefore, \(ch'(v)\geq-2+\frac{12}{25}+2=\frac{12}{25}\).
Second, assume that two faces which are incident with \(v\) are adjacent, say \(f_{1}\) and \(f_{2}\) are both 3-faces, then, \(f_{3}\) and \(f_{4}\) are both \(4^{+}\)-faces. If two false vertices that are adjacent to \(v\) are incident with the same \(4^{+}\)-face, without loss of generality assume that \(v_{1}\) and \(v_{4}\) are both false vertices, then, \(d(v_{i})\geq9\)\((i=2,3)\) by Lemma 2.1. So, \(v\) receives \(\frac{13}{50}\times2=\frac{13}{25}\) from \(v_{2}\) and \(v_{3}\) by R6, \(v\) receives at least \(\frac{2}{4}\times2=1\) from \(f_{3}\) and \(f_{4}\) by R1, therefore, \(ch'(v)\geq-2+\frac{12}{25}+\frac{13}{25}+1=0\). If two false vertices that are adjacent to \(v\) are incident with different \(4^{+}\)-faces, say \(f_{3}\) and \(f_{4}\), then, \(v_{1}\) and \(v_{3}\) are both false vertices, and \(d(v_{i})\geq9\)\((i=2,4)\) by Lemma 2.1. Since \(G\) has no 4-cycles, then, \(f_{3}\) and \(f_{4}\) are all \(5^{+}\)-faces. So, \(v\) receives at least \(\frac{4}{5-1}\times2=2\) from \(f_{3}\) and \(f_{4}\) by Lemma 2.5 and R1, Therefore, \(ch'(v)\geq-2+\frac{12}{25}+2=\frac{12}{25}\). If two false vertices that are adjacent to \(v\) are \(v_{2}\) and \(v_{4}\), since \(G\) has no 4-cycles, there is at least one \(5^{+}\)-face among \(f_{3}\) and \(f_{4}\), say \(f_{3}\). Thus, \(f_{3}\) sends at least \(\frac{4}{5-1}=1\) to \(v\), \(f_{4}\) sends at least \(\frac{2}{4-1}=\frac{2}{3}\) to \(v\) by Lemma 2.5 and R1. Therefore,\(ch'(v)\geq-2+\frac{12}{25}+1+\frac{2}{3}=\frac{11}{75}\).
Case 2.4. If \(v\) is adjacent to exactly three false vertices, say \(v_{1}\), \(v_{2}\) and \(v_{3}\), since \(G\) has exactly two 3-faces, so, \(f_{3}\) and \(f_{4}\) are all 3-faces by Lemma 2.4, \(f_{1}\) and \(f_{2}\) are all \(4^{+}\)-faces. Since \(G\) has not 4-cycles, so, \(f_{1}\) and \(f_{2}\) are either all 4-faces, or all \(5^{+}\)-faces, or there is at least one \(6^{+}\)-face. First assume that there is one \(6^{+}\)-face among \(f_{1}\) and \(f_{2}\), say \(f_{1}\). Let \(x_{i}\) \((i=1,2)\) be the second(undefined) neighbors of \(v_{2}\) on \(f_{i}\), it is easy to check that \(x_{1}x_{2}\)\(\in\)\(E(G)\) by the drawing of \(G\). Thus, at least one of \(x_{1}\) and \(x_{2}\) is big by Lemma 2.1. Then, \(v\) receives at least \(\min\{\frac{6}{6-1}+\frac{1}{2}, \frac{6}{6}+\frac{2}{4-1}\}\) \(=\frac{5}{3}\) from \(f_{1}\) and \(f_{2}\) by R1.
Therefore, \(ch'(v)\geq-2+\frac{12}{25}+\frac{5}{3}=\frac{11}{75}\). Second, assume that \(f_{1}\) and \(f_{2}\) are all \(5^{+}\)-faces, then, \(v\) receives at least \(\frac{4}{5-1}\times2=2\) from \(f_{1}\) and \(f_{2}\) by Lemma 2.5 and R1, thus, \(ch'(v)\geq-2+\frac{12}{25}+2=\frac{12}{25}\). Third, assume that \(f_{1}\) and \(f_{2}\) are all 4-faces, then, \(v_{2}\) is incident with four 4-faces, because otherwise, \(G\) has 4-cycles. By R8, \(v\) receives \(\frac{5}{12}\) from \(v_{2}\). Let \(x_{i}\) \((i=1,2)\) be the fourth(undefined) vertices of the 4-faces \(f_{i}\), it is easy to check that \(x_{1}x_{2}\)\(\in\)\(E(G)\) by the drawing of \(G\). Thus, at least one of \(x_{1}\) and \(x_{2}\) is big by Lemma 2.1. This implies that \(v\) receives at least \(\frac{1}{2}+\frac{2}{3}=\frac{7}{6}\) from \(f_{1}\) and \(f_{2}\) by R1. Therefore, \(ch'(v)\geq-2+\frac{12}{25}+\frac{5}{12}+\frac{7}{6}=\frac{19}{300}\)
Case 3. \(d=4\) and \(v\) is a false vertex, then, the neighbors of \(v\) are real vertices, and \(v\) is adjacent to at most two small vertices in \(G\) by Lemma 2.1. Since \(G\) has no 4-cycles, so, \(v\) is incident with at most two 3-faces, we consider three subcases.
Case 3.1. If \(v\) is not incident with any 3-face in \(G^{\times}\), then \(v\) is incident with four \(4^{+}\)-faces in \(G^{\times}\).
Assume first that \(v\) has at least one 4-neighbor, say \(v_{1}\), then, \(d(v_{3})\geq9\), moreover, there is at least one big among \(v_{2}\) and \(v_{4}\) by Lemma 2.1, say \(v_{2}\), thus, \(v\) would receive at least \(\frac{2}{4-2}=1\) from \(f_{2}\), at least \(\frac{2}{4-1}\times2=\frac{4}{3}\) from \(f_{1}\) and \(f_{3}\), at least \(\frac{2}{4}=\frac{1}{2}\) from \(f_{4}\) by R1, \(v\) sends at most \(\frac{5}{12}\times2=\frac{5}{6}\) to its 4-neighbors by R8. Therefore, \(ch'(v)\geq-2+1+\frac{4}{3}+\frac{1}{2}-\frac{5}{6}=0\). Otherwise, \(v\) does not have any 4-neighbors, then, \(v\) sends out nothing by R8, \(v\) would receive at least \(\frac{2}{4}\times4=2\), Thus,\(ch'(v)\geq-2+2=0\).
Case 3.2. If \(v\) is incident with exactly one 3-face in \(G^{\times}\), then without loss of generality assume that \(f_{1}\) is the 3-face. There is at least one big among \(v_{1}\) and \(v_{2}\), say \(v_{2}\).
Assume first that \(v_{1}\) is a 3-vertex, then, both \(v_{2}\) and \(v_{3}\) are \(10^{+}\)-vertices by Lemma 2.1. Thus, \(v\) would receive at least \(\frac{2}{4-2}=1\) from \(f_{2}\), at least \(\frac{2}{4-1}=\frac{2}{3}\) from \(f_{3}\), at least \(\frac{2}{4}=\frac{1}{2}\) from \(f_{4}\) by R1, \(v\) would receive \(\frac{13}{50}\) from \(v_{2}\) by R5, \(v\) sends at most \(\frac{1}{3}\) to \(v_{1}\) by R9. Thus, \(ch'(v)\geq-2+1+\frac{2}{3}+\frac{1}{2}+\frac{13}{50}-\frac{1}{3}=\frac{7}{75}\). Second, assume that \(4\leq\)\(d(v_{1})\)\(\leq7\), then, both \(v_{2}\) and \(v_{3}\) are \(6^{+}\)-vertices by Lemma 2.1. So, \(v\) would receive at least \(\frac{2}{4-2}=1\) from \(f_{2}\), at least \(\frac{2}{4-1}=\frac{2}{3}\) from \(f_{3}\), at least \(\frac{2}{4}=\frac{1}{2}\) from \(f_{4}\) by R1, \(v\) sends out nothing by R9. Therefore,\(ch'(v)\geq-2+1+\frac{2}{3}+\frac{1}{2}=\frac{1}{6}\). Third, assume that \(v_{1}\) is a \(8^{+}\)-vertex, then, \(v_{1}\) sends \(\frac{13}{50}\) to \(v\) by R5. There is at least a big vertex among \(v_{2}\) and \(v_{4}\), so, \(f_{2}\) and \(f_{4}\) send \(\min\{{\frac{2}{4-2}+\frac{1}{2}, \frac{2}{4-1}\times2}\}\) \(=\frac{4}{3}\) to \(v\), \(f_{3}\) sends \(\frac{2}{4}=\frac{1}{2}\) to \(v\) by R1, \(v\) sends out nothing by R9. Therefore,\(ch'(v)\geq-2+\frac{13}{50}+\frac{4}{3}+\frac{1}{2}=\frac{7}{75}\).
Case 3.3. If \(v\) is incident with two 3-faces in \(G^{\times}\), since \(G\) has no 4-cycles, then, the two 3-faces have a common edge, without loss of generality assume that \(f_{3}\) and \(f_{4}\) are 3-faces. There is a big vertex among \(v_{1}\) and \(v_{3}\), say \(v_{1}\).
First assume that \(6\leq\)\(d(v_{4})\)\(\leq7\) , then, both \(v_{2}\) and \(v_{3}\) are big by Lemma 2.1, moreover, \(f_{1}\) and \(f_{2}\) send at least \(\frac{2}{4-2}\times2=2\) to \(v\) by R1, \(v\) sends out nothing by R7. Thus, \(ch'(v)\geq-2+2=0\). Second, assume that \(d(v_{4})\leq5\), then, \(v_{i}\) \((i=1,2,3)\) is \(8^{+}\)-vertex by Lemma 2.1, moreover, \(f_{1}\) and \(f_{2}\) send at least \(\frac{2}{4-2}\times2=2\) to \(v\) by R1, \(v_{1}\) and \(v_{3}\) send \(\frac{13}{50}\times2=\frac{13}{25}\) to \(v\) by R5, \(v\) sends at most \(\frac{13}{25}\) to \(v_{4}\) by R7. Thus,\(ch'(v)\geq-2+2+\frac{13}{25}-\frac{13}{25}=0\).
Third assume that \(v_{4}\) is a \(8^{+}\)-vertex. If there is at least one \(5^{+}\)-face among \(f_{1}\) and \(f_{2}\), say \(f_{1}\), then, \(f_{1}\) sends at least \(\frac{4}{5-1}=1\) to \(v\), \(f_{2}\) sends at least \(\frac{2}{4}=\frac{1}{2}\) to \(v\) by Lemma 2.5 and R1, \(v_{4}\) sends \(\frac{13}{50}\times2=\frac{13}{25}\) to \(v\) by R5. Thus, \(ch'(v)\geq-2+1+\frac{1}{2}+\frac{13}{25}=\frac{1}{50}\). Otherwise, \(f_{1}\) and \(f_{2}\) are all 4-faces. Let \(x_{i}\)\((i=1,2)\) be the second(undefined) neighbors of \(v_{2}\) on \(f_{i}\), since \(G\) has no 4-cycles, then, both \(x_{1}\) and \(x_{2}\) are false vertices. Suppose that \(v_{2}\) is a \(6^{+}\)-vertex, then, \(f_{1}\) sends at least \(\frac{2}{4-2}=1\) to \(v\), \(f_{2}\) sends at least \(\frac{2}{4-1}=\frac{2}{3}\) to \(v\) by R1, \(v_{4}\) sends \(\frac{13}{50}\times2=\frac{13}{25}\) to \(v\) by R5. Thus, \(ch'(v)\geq-2+1+\frac{2}{3}+\frac{13}{25}=\frac{14}{75}\). Suppose that \(v_{2}\) is a 3-vertex or a 4-vertex, since \(G\) has no 4-cycles, then, \(v_{2}\) is not incident with any 3-faces. By R10 and R11, \(v_{2}\) sends at least \(\frac{11}{75}\) to \(v\). Suppose that \(v_{2}\) is a 5-vertex, by R12,\(v_{2}\) sends \(\frac{1}{6}\) to \(v\). Thus, when \(v_{2}\) is a \(5^{-}\)-vertex, this implies that \(v_{2}\) sends at least \(\frac{11}{75}\) to \(v\). And moreover, we consider the degree of \(v_{3}\). If \(v_{3}\) is a \(5^{-}\)-vertex, then, \(v_{1}\) is a \(8^{+}\)-vertex by Lemma 2.1, \(v_{1}\) sends \(\frac{13}{50}\) to \(v\), \(v_{4}\) sends \(\frac{13}{50}\times2=\frac{13}{25}\) to \(v\) by R5, \(f_{1}\) sends at least \(\frac{2}{4-1}=\frac{2}{3}\) to \(v\), \(f_{2}\) sends at least \(\frac{2}{4}=\frac{1}{2}\) to \(v\) by R1. Therefore, \(ch'(v)\geq-2+\frac{11}{75}+\frac{13}{50}+\frac{13}{25}+\frac{2}{3}+\frac{1}{2}=\frac{7}{75}\). If \(v_{3}\) is a \(6^{+}\)-vertex, since \(v_{1}\) is big, then, each of \(f_{1}\) and \(f_{2}\) sends at least \(\frac{2}{4-1}=\frac{2}{3}\) to \(v\) by R1, \(v_{4}\) sends \(\frac{13}{50}\times2=\frac{13}{25}\) to \(v\) by R5, therefore, \(ch'(v)\geq-2+\frac{11}{75}+\frac{2}{3}\times2+\frac{13}{25}=0\).
Case 4. \(d=5\). \(v\) is incident with at most four 3-faces in \(G^{\times}\) by Lemma 2.3. If \(v\) would send charges to a false vertex which adjacent to \(v\) by R12, then \(v\) is incident with at most three 3-faces in \(G^{\times}\). First assume that \(v\) is incident with exactly four 3-faces in \(G^{\times}\), say \(f_{1}\), \(f_{2}\), \(f_{3}\) and \(f_{4}\), then \(v_{1}\),\(v_{3}\), and \(v_{5}\) are false vertices, \(v_{2}\) and \(v_{4}\) are real vertices by Lemma 2.8. Since \(G\) has no 4-cycles, then, \(f_{5}\) is a \(6^{+}\)-face. Thus, \(f_{5}\) sends 1 to \(v\) by R1, therefore, \(ch'(v)\geq-1+1=0\). Second assume that \(v\) is incident with exactly three 3-faces in \(G^{\times}\), then, \(v\) is incident with two \(4^{+}\)-faces. If the two \(4^{+}\)-faces are not adjacent, then, \(v\) sends out nothing by R12, \(v\) would receive at least \(\frac{2}{4}\times2=1\) from two \(4^{+}\)-faces which are incident with \(v\). Thus, \(ch'(v)\geq-1+1=0\). If two \(4^{+}\)-faces are adjacent, without loss of generality, assume that \(f_{1}\) and \(f_{2}\) are \(4^{+}\)-faces. Moreover, if \(v_{2}\) is a real vertex, then, \(v\) sends out nothing by R12, \(v\) would receive at least \(\frac{2}{4}\times2=1\) from \(f_{1}\) and \(f_{2}\) by R1. Thus, \(ch'(v)\geq-1+1=0\). Otherwise, \(v_{2}\) is a false vertex, then, \(v\) sends at most \(\frac{1}{6}\) to \(v_{2}\) by R12. Let \(x_{i}\) be the second(undefined) neighbors of \(v_{2}\) on \(f_{i}\) \((i=1,2)\), it is easy to check that \(x_{1}x_{2}\)\(\in\)\(E(G)\) by the drawing of \(G\). Thus, at least one of \(x_{1}\) and \(x_{2}\) is big by Lemma 2.1. This implies \(v\) would receive at least \(\frac{2}{4}+\frac{2}{4-1}=\frac{7}{6}\) from \(f_{1}\) and \(f_{2}\) by R1, therefore, \(ch'(v)\geq-1+\frac{7}{6}-\frac{1}{6}=0\). Third assume that \(v\) is incident with at most two 3-faces in \(G^{\times}\), then, \(v\) is incident with at least three \(4^{+}\)-faces. \(v\) sends at most \(\frac{1}{6}\times3=\frac{1}{2}\) to false vertices which are adjacent to \(v\) by R12. \(v\) would receive at least \(\frac{2}{4}\times3=\frac{3}{2}\) from \(4^{+}\)-faces which are incident with \(v\), thus, \(ch'(v)\geq-1+\frac{3}{2}-\frac{1}{2}=0\).
Case 5. \(6\leq d\leq 7\). Then it is trivial that \(ch'(v)=ch(v)\geq0\).
Case 6. \(8\leq d\leq \Delta(G)-2\). By Lemma 2.3, \(v\) is incident with at most \(\lfloor\frac{4}{5}d\rfloor\) 3-faces in \(G^{\times}\), then \(v\) sends at most \(\lfloor\frac{4}{5}d\rfloor\times\frac{13}{50}\) to false vertices and real 4-vertices which are adjacent to \(v\) on 3-faces by R5 and R6. Thus, \(ch'(v)\geq d-6-\lfloor\frac{4}{5}d\rfloor\times\frac{13}{50}\geq\frac{99d-750}{125}>0\), since \(d\geq8\).
Case 7. \(d=\Delta(G)-1\). By Lemma 2.3, \(v\) is incident with at most \(\lfloor\frac{4}{5}d\rfloor\) 3-faces in \(G^{\times}\). And by Lemma 2.1, we have \(d(u)\geq4\) if \(uv\in E(G)\). So \(v\) can be a 4-master vertex of at most two vertices and a 5-master vertex of at most three vertices by Lemma 2.6.
Let \(\Delta(G)=10\), Then, \(d=9\). By Lemma 2.3, \(v\) is incident with at most seven 3-faces in \(G^{\times}\). If \(v\) is incident with exactly seven 3-faces in \(G^{\times}\), then, by Lemma 2.8, there are four consecutive 3-faces and another three consecutive 3-faces which are incident with \(v\), and \(v\) is adjacent to at most two real small vertices. Thus, \(v\) sends at most \(7\times\frac{13}{50}\) to false vertices and real 4-vertices which are adjacent to \(v\) by R5 and R6. \(v\) sends at most \(\frac{6}{25}\times2=\frac{12}{25}\) by R3. Therefore, \(ch'(v)\geq 9-6-7\times\frac{13}{50}-\frac{12}{25}=\frac{7}{10}\). If \(v\) is incident with at most six 3-faces in \(G^{\times}\), then, \(v\) sends at most \(6\times\frac{13}{50}\) to false vertices and real 4-vertices which are adjacent to \(v\) by R5 and R6, \(v\) sends at most \(\frac{6}{25}\times2+\frac{6}{25}\times3=\frac{6}{5}\) to real small vertices which are adjacent to it by R3. Thus, \(ch'(v)\geq 9-6-6\times\frac{13}{50}-\frac{6}{5}=\frac{6}{25}\).
Let \(\Delta(G)\geq11\), then, \(d\geq10\). By Lemma 2.3, \(v\) is incident with at most \(\lfloor\frac{4}{5}d\rfloor\) 3-faces in \(G^{\times}\). \(v\) sends at most \(\lfloor\frac{4}{5}d\rfloor\times\frac{13}{50}\) to false vertices and real 4-vertices which are adjacent to \(v\) by R5 and R6, \(v\) sends out at most \(\frac{6}{25}\times2+\frac{6}{25}\times3=\frac{6}{5}\) by R3. Thus \(ch'(v)\geq \Delta(G)-1-6-\lfloor\frac{4}{5}(\Delta(G)-1)\rfloor\times\frac{13}{50}-\frac{6}{5}\geq\frac{99\Delta(G)-999}{125}\geq0\), since \(\Delta(G)\geq11\).
Case 8. \(d=\Delta(G)\). By Lemma 2.3, \(v\) is incident with at most \(\lfloor\frac{4}{5}d\rfloor\) 3-faces in \(G^{\times}\). And by Lemma 2.1, we have \(d(u)\geq3\) if \(uv\in E(G)\). So \(v\) can be a 3-master vertex of at most one vertex, a 4-master vertex of at most two vertices and a 5-master vertex of at most three vertices by Lemma 2.6.
If \(d=10\), then \(v\) is incident with at most eight 3-faces. When \(v\) is incident with exactly eight 3-faces, there are two groups four consecutive 3-faces that are incident with \(v\) in \(G^{\times}\), so, \(v\) is adjacent to at most two real small vertices by Lemma 2.8. Thus, \(v\) sends at most \(\frac{6}{25}\times2=\frac{12}{25}\) to real small vertices that are adjacent to it by R2 and R3, \(v\) sends at most \(8\times\frac{13}{50}\) to false vertices and real 4-vertices that are adjacent to \(v\) by R5 and R6, and \(\frac{1}{2}\) to the common pot by R4. Therefore, \(ch'(v)\geq10-6-8\times\frac{13}{50}-\frac{12}{25}-\frac{1}{2}=\frac{47}{50}\geq0\). When \(v\) is incident with at most seven 3-faces, \(v\) sends at most \(7\times\frac{13}{50}\) to false vertices and real 4-vertices that are adjacent to \(v\) by R5 and R6, \(v\) sends at most \(\frac{6}{25}\times5+\frac{2}{9}=\frac{64}{45}\) to real small vertices that are adjacent to it by R2 and R3, \(v\) sends \(\frac{1}{2}\) to the common pot by R4. Therefore, \(ch'(v)\geq10-6-7\times\frac{13}{50}-\frac{64}{45}-\frac{1}{2}=\frac{58}{225}\geq0\).
If \(d\geq11\), \(v\) sends at most \(\lfloor\frac{4}{5}d\rfloor\times\frac{13}{50}\) to false vertices and real 4-vertices that are adjacent to \(v\) by R5 and R6, \(v\) sends at most \(\frac{6}{25}\times5+\frac{2}{9}=\frac{64}{45}\) to real small vertices adjacent to it by R2, R3 and \(\frac{1}{2}\) to the common pot by R4. Thus \(ch'(v)\geq \Delta(G)-6-\lfloor\frac{4}{5}\Delta(G)\rfloor\times\frac{13}{50}-\frac{64}{45}-\frac{1}{2}\geq\frac{1782\Delta(G)-17825}{2250}\geq0\), since \(\Delta(G)\geq11\).
Therefore, we complete the proof of the Theorem.