Let \(R(a(x-y) = bz)\) denote the least integer \(n\) such that for every \(2\)-coloring of the set \(\{1, 2, \ldots, n\}\) there exists a monochromatic solution to \(a(x-y) = bz\). Recently, Gasarch, Moriarty, and Tumma conjectured that \(R(a(x-y) = bz) = b^2 + b + 1\), where \(1 < a < b\). In this note, we confirm this conjecture.

In this paper, we introduce the notion of a generalized triple derivation \(f\), with an associated triple derivation \(d\), on a lattice and investigate some related results. Among some other results, we prove that: Let \((L, \wedge, \vee)\) be a distributive lattice and \(f\) be a generalized triple derivation, with associated triple derivation \(d\), on \(L\). Then the following conditions are equivalent for all \(x, y, z \in L\):

1. \(f\) is an isotone generalized triple derivation on \(L\),
2. \(f_{x \wedge y \wedge z} = f_x \wedge f_y \wedge f_z\),
3. \(f_{x \vee y \vee z} = f_x \vee f_y \vee f_z\).

The scrambling index of an \(n \times n\) primitive matrix \(A\) is the smallest positive integer \(k\) such that \(A^k(A^T)^k > 0\), where \(A^T\) denotes the transpose of \(A\). In 2009, M. Akelbek and S. Kirkland gave an upper bound on the scrambling index of an \(n \times n\) primitive matrix \(M\) in terms of its order \(n\), and they also characterized the primitive matrices that achieve the upper bound. In this paper, we characterize primitive matrices which achieve the second largest scrambling index in terms of its order. Meanwhile, we show that there exists a gap in the scrambling index set of primitive matrices.

Let \(d_G(v)\) be the degree of a vertex \(v\) in a graph \(G\). A graph \(G\) is called a \(D(i_1, \ldots,i_k)\) graph, if \(\{d_G(v) \mid x \in V(G)\} = \{i_1, \ldots, i_k\}\). In this paper, a necessary and sufficient condition for a connected \(D(1, 3)\) graph to be cordial is given.

Let \(G\) be a connected graph of order \(n\), and suppose that \(n = \sum_{i=1}^{k}n_i\), where \(n_1, n_2, \ldots,n_n\) are integers with at least two. A spanning subgraph is called a path-factor if each component of it is a path of order at least two. In [Y. Chen, F. Tian, B, Wei, Degree sums and path-factors in graphs, Graphs and Combin. \(17 (2001),61-71.]\), Chen et al. gave a degree sum condition for the existence of a path-factor consisting of paths of order \(n_1, n_2, \ldots, n_k\). In this paper, for 2-connected graphs, we generalize this result.

Let \(G\) be a graph with \(n\) vertices and \(\mu_1, \mu_2, \ldots, \mu_n\) be the Laplacian eigenvalues of \(G\). The Laplacian-energy-like graph invariant \(\text{LEL}(G) = \sum_{i=1}^{n} \sqrt{\mu_i}\) has been defined and investigated in [1]. Two non-isomorphic graphs \(G_1\) and \(G_2\) of the same order are said to be \(\text{LEL}\)-equienergetic if \(\text{LEL}(G_1) = \text{LEL}(G_2)\). In [2], three pairs of \(\text{LEL}\)-equienergetic non-cospectral connected graphs are given. It is also claimed that the \(\text{LEL}\)-equienergetic non-cospectral connected graphs are relatively rare. It is natural to consider the question: Whether the number of the \(\text{LEL}\)-equienergetic non-cospectral connected graphs is finite? The answer is negative, because we shall construct a pair of \(\text{LEL}\)-equienergetic non-cospectral connected graphs of order \(n\), for all \(n \geq 12\) in this paper.

The status of a vertex \(v\) in a graph is the sum of the distances between \(v\) and all vertices. The status sequence of a graph is the list of the statuses of all vertices arranged in nondecreasing order. It is well known that non-isomorphic graphs may have the same status sequence. This paper gives a sufficient condition for a graph \(G\) with the property that there exists another graph \(G’\) such that \(G’\) and \(G\) have the same status sequence and \(G’\) is not isomorphic to \(G\).

We give combinatorial proofs of some binomial and $q$-binomial identities in the literature, such as

\[\sum\limits_{k={-\infty}}^{\infty}(-1)^kq^{\frac{(9k^2+3k)}{2}}\binom{2n}{n+3k}=(1+q^n)\prod\limits_{k=1}^{n-1}(1+q^k+q^{2k})(n\geq 1)\]

and

\[\sum\limits_{k=0}^{\infty} \binom{3n}{2k}(-3)^k=(-8)^n.\]

Two related conjectures are proposed at the end of this paper.

In the spirit of Ryser’s theorem, we prove sufficient conditions on \(k\), \(\ell\), and \(m\) so that \(k \times \ell \times m\) Latin boxes, i.e., partial Latin cubes whose filled cells form a \(k \times \ell \times m\) rectangular box, can be extended to a \(k \times n \times m\) Latin box, and also to a \(k \times n \times m\) Latin box, where \(n\) is the number of symbols used, and likewise the order of the Latin cube. We also prove a partial Evans-type result for Latin cubes, namely that any partial Latin cube of order \(n\) with at most \(n-1\) filled cells is completable, given certain conditions on the spatial distribution of the filled cells.

A star-factor of a graph \(G\) is a spanning subgraph of \(G\) such that each component is a star. An edge-weighting of \(G\) is a function \(w: E(G) \rightarrow \mathbb{N}^+\), where \(\mathbb{N}^+\) is the set of positive integers. Let \(\Omega\) be the family of all graphs \(G\) such that every star-factor of \(G\) has the same weight under some fixed edge-weighting \(w\). The open problem of characterizing the class \(\Omega\), posed by Hartnell and Rall, is motivated by the minimum cost spanning tree and the optimal assignment problems. In this paper, we present a simple structural characterization of the graphs in \(\Omega\) that have girth at least five.

We show that whenever the length four words over a three letter alphabet are two-colored, there must exist a monochromatic combinatorial line. We also provide some computer generated lower bounds for some other Hales-Jewett numbers.

This paper introduces a method for finding closed forms for certain sums involving squares of binomial coefficients. We use this method to present an alternative approach to a problem of evaluating a different type of sums containing squares of the numbers from
Catalan’s triangle.

In this paper, we deal with a special kind of hypergraph decomposition. We show that there exists a decomposition of the 3-uniform hypergraph \(\lambda K_v^{(3)}\) into a special kind of hypergraph \(K_{4}^{(3)} – e\) whose leave has at most two edges, for any positive integers \(v \geq 4 \) and \(\lambda\).

For a connected graph \(G\) of order \(n \geq 2\) and a linear ordering \(s = v_1, v_2, \ldots, v_n\) of \(V(G)\), define \(d(s) = \sum_{i=1}^{n-1} d(v_i, v_{i+1})\), where \(d(v_i, v_{i+1})\) is the distance between \(v_i\) and \(v_{i+1}\). The traceable number \(t(G)\) and upper traceable number \(t^+(G)\) of \(G\) are defined by \(t(G) = \min\{d(s)\}\) and \(t^+(G) = \max\{d(s)\}\), respectively, where the minimum and maximum are taken over all linear orderings \(s\) of \(V(G)\). The traceable number \(t(v)\) of a vertex \(v\) in \(G\) is defined by \(t(v) = \min\{d(s)\}\), where the minimum is taken over all linear orderings \(s\) of \(V(G)\) whose first term is \(v\). The \({maximum\; traceable \;number}\) \(t^*(G)\) of \(G\) is then defined by \(t^*(G) = \max\{t(v) : v \in V(G)\}\). Therefore, \(t(G) \leq t^*(G) \leq t^+(G)\) for every nontrivial connected graph \(G\). We show that \(t^*(G) \leq \lfloor \frac{t(G)+t^+(G)+1}{2}\rfloor\) for every nontrivial connected graph \(G\) and that this bound is sharp. Furthermore, it is shown that for positive integers \(a\) and \(b\), there exists a nontrivial connected graph \(G\) with \(t(G) = a\) and \(t^*(G) = b\) if and only if \(a \leq b \leq \left\lfloor \frac{3n}{2} \right\rfloor\).

Let \(G\) be a simple graph with \(n\) vertices and \(m\) edges, and let \(\lambda_1\) and \(\lambda_2\) denote the largest and second largest eigenvalues of \(G\). For a nontrivial bipartite graph \(G\), we prove that:
(i) \(\lambda_1 \leq \sqrt{m – \frac{3-\sqrt{5}}{2}}\), where equality holds if and only if \(G \cong P_4\);
(ii) If \(G \ncong P_n\), then \(\lambda_1 \leq \sqrt{{m} – (\frac{5-\sqrt{17}}{2})}\), where equality holds if and only if \(G \cong K_{3,3} – e\);
(iii) If \(G\) is connected, then \(\lambda_2 \leq \sqrt{{m} – 4{\cos}^2(\frac{\pi}{n+1})}\), where equality holds if and only if \(G \cong P_{n,2} \leq n \leq 5\);
(iv) \(\lambda_2 \geq \frac{\sqrt{5}-1}{2}\), where equality holds if and only if \(G \cong P_4\);
(v) If \(G\) is connected and \(G \ncong P_n\), then \(\lambda_2 \geq \frac{5-\sqrt{17}}{2}\), where equality holds if and only if \(G \cong K_{3,3} – e\).

Let \(n\) be a positive integer. Denote by \(PG(n,q)\) the \(n\)-dimensional projective space over the finite field \(\mathbb{F}_q\) of order \(q\). A blocking set in \(PG(n,q)\) is a set of points that has non-empty intersection with every hyperplane of \(PG(n,q)\). A blocking set is called minimal if none of its proper subsets are blocking sets. In this note, we prove that if \(PG(n_i,q)\) contains a minimal blocking set of size \(k_i\) for \(i \in \{1,2\}\), then \(PG(n_1 + n_2 + 1,q)\) contains a minimal blocking set of size \(k_1 + k_2 – 1\). This result is proved by a result on groups with maximal irredundant covers.

A graph is said to be edge-transitive if its automorphism group acts transitively on its edge set. In this paper, all connected cubic edge-transitive graphs of order \(12p\) or \(12p^2\) are classified.

For any \(n\geq 7\), we prove that there exists a tournament of order \(n\), such that for each pair of distinct vertices there exists a path of length \(2\).

A \((k, t)\)-list assignment \(L\) of a graph \(G\) assigns a list of \(k\) colors available at each vertex \(v\) in \(G\) and \(|\bigcup_{v\in V(G)}L(v)| = t\). An \(L\)-coloring is a proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\). A graph \(G\) is \((k,t)\)-choosable if \(G\) has an \(L\)-coloring for every \((k, t)\)-list assignment \(L\).
Erdős, Rubin, and Taylor proved that a graph is \((2, t)\)-choosable for any \(t > 2\) if and only if a graph does not contain some certain subgraphs. Chareonpanitseri, Punnim, and Uiyyasathian proved that an \(n\)-vertex graph is \((2,t)\)-choosable for \(2n – 6 \leq t \leq 2n – 4\) if and only if it is triangle-free. Furthermore, they proved that a triangle-free graph with \(n\) vertices is \((2, 2n – 7)\)-choosable if and only if it does not contain \(K_{3,3} – e\) where \(e\) is an edge. Nakprasit and Ruksasakchai proved that an \(n\)-vertex graph \(G\) that does not contain \(C_5 \vee K_{n-2}\) and \(K_{4,4}\) for \(k \geq 3\) is \((k, kn – k^2 – 2k)\)-choosable. For a non-2-choosable graph \(G\), we find the minimum \(t_1 \geq 2\) and the maximum \(t_2\) such that the graph \(G\) is not \((2, t_i)\)-choosable for \(i = 1, 2\) in terms of certain subgraphs. The results can be applied to characterize \((2, t)\)-choosable graphs for any \(t\).

Let \(G\) be the circuit graph of any connected matroid. It is proved that the circuit graph of a connected matroid with at least three circuits is \(E_2\)-Hamiltonian.

The Randić index \(R(G)\) of a graph \(G\) is defined by \(R(G) = \sum\limits_{uv} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) is the degree of a vertex \(u\) in \(G\) and the summation extends over all edges \(uv\) of \(G\). In this work, we give sharp lower bounds of \(R(G) + g(G)\) and \(R(G) . g(G)\) among \(n\)-vertex connected triangle-free graphs with Randić index \(R\) and girth \(g\).

Hammack and Livesay introduced a new graph operation \(G^{(k)}\) for a graph \(G\), which they called the \(k\)th inner power of \(G\). A graph \(G\) is Hamiltonian if it contains a spanning cycle. In this paper, we show that \(C^{(k)}_n(n \geq 3, k \geq 2)\) is Hamiltonian if and only if \(n\) is odd and \(k = 2\), where \(C_n\) is the cycle with \(n\) vertices.

Let \(a(v)\) and \(g(v)\) denote the least possible area and the least possible number of lattice points in the interior of a convex lattice \(v\)-gon, respectively. Many lower and upper bounds for \(a(v)\) and \(g(v)\) are known for every \(v\). However, the exact values of these two functions are only known for \(v \leq 10\) and \(v \in \{12, 13, 14, 16, 18, 20, 22\}\). The purpose of this paper is to answer the following Open Question 1 from \([13]\): What is the exact value of \(a(11)\)? We answer this question by proving that \(a(11) = 21.5\). On our way to achieve this goal, we also prove that \(g(11) = 17\).

The edge-face total chromatic number of \(3\)-regular Halin graphs was shown to be \(4\) or \(5\) in \([5]\). In this paper, we shall provide a necessary and sufficient condition to characterize \(3\)-regular Halin graphs with edge-face total chromatic number equal to four.

Jaeger \(et \;al\). [ J. Combin. Theory, Ser B, \(56 (1992) 165-182]\) conjectured that every 3-edge-connected graph is \(Z_5\)-connected. Let \(G\) be a 3-edge-connected simple graph on \(n\) vertices and \(A\) an abelian group with \(|A| \geq 3\). If a graph \(G^*\) is obtained by repeatedly contracting nontrivial \(A\)-connected subgraphs of \(G\) until no such subgraph is left, we say \(G\) can be \(A\)-reduced to \(G^*\). It is proved in this paper that \(G\) is \(A\)-connected with \(|A| \geq 5\) if one of the following holds: (i) \(n \leq 15\); (ii) \(n = 16\) and \(\Delta \geq 4\); or (iii) \(n = 17\) and \(\Delta \geq 5\). As applications, we also show the following results:
(1) For \(|A| \geq 5\) and \(n \geq 17\), if \(|E(G)| \geq \binom{n-15}{2} + 31\), then \(G\) is \(A\)-connected.
(2) For \(|A| \geq 4\) and \(n \geq 13\), if \(|E(G)| \geq \binom{n-11}{2} + 23\), then either \(G\) is \(A\)-connected or \(G\) can be \(A\)-reduced to the Petersen graph.

Given a partial cube \(G\), the \(\Theta\)-graph of \(G\) has \(\Theta\)-classes of \(G\) as its vertices, and two vertices in it are adjacent if the corresponding \(\Theta\)-classes meet in a vertex of \(G\). We present a counter-example to the question from \([8]\) whether \(\Theta\)-graphs of graphs of acyclic cubical complexes are always dually chordal graphs. On a positive side, we show that in the class of ACC \(p\)-expansion graphs, each \(\Theta\)-graph is both a dually chordal and a chordal graph. In the proof, a fundamental characterization of \(\Theta\)-acyclic hypergraphs is combined with techniques from metric graph theory. Along the way, we also introduce a new, weaker version of simplicial elimination scheme, which yields yet another characterization of chordal graphs.

Let \(X = (V, E)\) be a connected vertex-transitive graph with degree \(k\). Call \(X\) super restricted edge-connected, in short, sup-\(\lambda’\), if \(F\) is a minimum edge set of \(X\) such that \(X – F\) is disconnected and every component of \(X – F\) has at least two vertices, then \(F\) is the set of edges adjacent to a certain edge in \(X\). Wang [Y, Q, Wang, Super restricted edge-connectivity of vertex-transitive graphs, Discrete Mathematics \(289 (2004) 199-205]\) proved that a connected vertex-transitive graph with degree \(k > 2\) and girth \(g > 4\) is sup-\(\lambda’\). In this paper, by studying the \(k\)-superatom of \(X\), we present sufficient and necessary conditions for connected vertex-transitive graphs and Cayley graphs with degree \(k > 2\) to be sup-\(\lambda’\). In particular, sup-\(\lambda’\) connected vertex-transitive graphs with degree \(k > 2\) and girth \(g > 3\) are completely characterized. These results can be seen as an improvement of the one obtained by Wang.

A proper vertex coloring of a graph \(G\) is called a dynamic coloring if for every vertex \(v\) with degree at least 2, the neighbors of \(v\) receive at least two different colors. It was conjectured that if \(G\) is a regular graph, then \(\chi_2(G) – \chi(G) \leq 2\). In this paper, we prove that, apart from the cycles \(C_4\) and \(C_5\) and the complete bipartite graphs \(K_{n,n}\), every strongly regular graph \(G\) satisfies \(\chi_2(G) – \chi(G) \leq 1\).

Let \(\vec{P_l}\) be the directed path on \(r\) vertices and \(\lambda K^*_{m,n}\) be the symmetric complete bipartite multi-digraph with two partite sets having \(m\) and \(n\) vertices. A \(\vec{P_l}\)-factorization of \(\lambda K^*_{m,n}\) is a set of arc-disjoint \(\vec{P_l}\)-factors of \(\lambda K^*_{m,n}\), which is a partition of the set of arcs of \(\lambda K^*_{m,n}\). In this paper, it is shown that a necessary and sufficient condition for the existence of a \(\vec{P}_{2k+l}\)-factorization of \(\lambda K^*_{m,n}\) for any positive integer \(k\).

Let \(G = (V, E)\) be a finite non-empty graph. A vertex-magic total labeling (VMTL) is a bijection \(\lambda\) from \(V \cup E\) to the set of consecutive integers \(\{1, 2, \ldots, |V| + |E|\}\) with the property that for every \(v \in V\), \(\lambda(v) + \sum_{w \in N(v)} \lambda(vw) = h\), for some constant \(h\). Such a labeling is called super if the vertex labels are \(1, 2, \ldots, |V|\).
There are some results known about super VMTLs of \(kG\) only when the graph \(G\) has a super VMTL. In this paper, we focus on the case when \(G\) is the complete graph \(K_n\). It was shown that a super VMTL of \(kK_n\) exists for \(n\) odd and any \(k\), for \(4 < n \equiv 0 \pmod{4}\) and any \(k\), and for \(n = 4\) and \(k\) even. We continue the study and examine the graph \(kK_n\) for \(n \equiv 2 \pmod{4}\). Let \(n = 4l + 2\) for a positive integer \(l\). The graph \(kK_{4l+2}\) does not admit a super VMTL for \(k\) odd. We give a large number of super VMTLs of \(kK_{4l+2}\) for any even \(k\) based on super VMTLs of \(4K_{2l+1}\).

For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph. Let \(K_m – H\) be the graph obtained from \(K_m\) by removing the edge set \(E(H)\), where \(H\) is a subgraph of \(K_m\). In this paper, we characterize the potentially \(K_6 – C_4\)-graphic sequences. This characterization implies a theorem due to Hu and Lai \([7]\).

Double Fibonacci sequences \((x_{n,k})\) are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.

A set \(S \subseteq V\) is a dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is either in \(S\) or is adjacent to a vertex in \(S\). A vertex is said to dominate itself and all its neighbors. The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). In terms of a chess board problem, let \(X_n\) be the graph for chess piece \(X\) on the square of side \(n\). Thus, \(\gamma(X_n)\) is the domination number for chess piece \(X\) on the square of side \(n\). In 1964, Yaglom and Yaglom established that \(\gamma(K_n) = \left\lceil \frac{n+2}{2} \right\rceil^2\). This extends to \(\gamma(K_{m,n}) = \left\lceil \frac{m+2}{3} \right\rceil \left\lceil \frac{n+2}{3} \right\rceil\) for the rectangular board. A set \(S \subseteq V\) is a total dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is adjacent to a vertex in \(S\). A vertex is said to dominate its neighbors but not itself. The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). In 1995, Garnick and Nieuwejaar conducted an analysis of the total domination numbers for the king’s graph on the \(m \times n\) board. In this paper, we note an error in one portion of their analysis and provide a correct general upper bound for \(\gamma_t(K_{m,n})\). Furthermore, we state improved upper bounds for \(\gamma_t(K_n)\).

A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually the positive integers). An \((a, d)\)-edge-antimagic total labeling of a graph with \(p\) vertices and \(q\) edges is a one-to-one mapping that takes the vertices and edges onto the integers \(1, 2, \ldots, p + q\), such that the sums of the label on the edges and the labels of their end points form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called \({super}\) if the smallest possible labels appear on the vertices. In this paper, we study the super \((a, 2)\)-edge-antimagic total labelings of disconnected graphs. We also present some necessary conditions for the existence of \((a, d)\)-edge-antimagic total labelings for \(d\) even.

Fault tolerance is an important property of network performance. A graph \(G\) is \(k\)-edge-fault conditional Hamiltonian if \(G – F\) is Hamiltonian for every \(F \subset E(G)\) with \(|F| \leq k\) and \(\delta(G – F) \geq 2\). In this paper, we show that for \(n \geq 4\), the \(n\)-dimensional star graph \(S_n\) is \((3n – 10)\)-edge-fault conditional Hamiltonian.

In this paper, we characterize all spacelike, timelike, and null curves lying on the pseudohyperbolic space \({H}^{4}_{v-1}\), in Minkowski space \({E}^5_v\). Moreover, we prove that there are no timelike and no null curves lying on the pseudohyperbolic space \({H}^{4}_{v-1}\) in \({E}^5_v\).

In 1982, Beutelspacher and Brestovansky proved that for every integer \(m \geq 3\), the \(2\)-color Rado number of the equation
\[x_1+x_2+ \ldots + x_{m-1}=x_m\]
is \(m^2 – m – 1\). In 2008, Schaal and Vestal proved that, for every \(m \geq 6\), the \(2\)-color Rado number of
\[x_1+x_2+ \ldots + x_{m-1}=2x_m\]
is \(\left\lceil \frac{m-1}{2}\left\lceil \frac{m-1}{2} \right\rceil \right\rceil \). Here, we prove that, for every integer \(a \geq 3\) and every \(m \geq 2a^2 – a + 2\), the 2-color Rado number of
\[x_1+x_2+ \ldots + x_{m-1}=ax_m\]
is \(\left\lceil \frac{m-1}{a}\left\lceil \frac{m-1}{a} \right\rceil \right\rceil\). For the case \(a = 3\), we show that our formula gives the Rado number for all \(m \geq 7\), and we determine the Rado number for all \(m \geq 3\).

The general Randic index \(R_{-\alpha}(G)\) of a graph \(G\), defined by a real number \(\alpha\), is the sum of \((d(u)d(v))^{-\alpha}\) over all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we have discussed some properties of the Max Tree which has the maximum general Randic index \(R_{-\alpha}(G)\), where \(\alpha \in (\alpha_0,2)\). Based on these properties, we are able to obtain the structure of the Max Tree among all trees of order \(k \geq 3\). Thus, the maximal value of \(R_{-\alpha}(G)\) follows easily.

A \(\lambda\)-fold \(G\)-design of order \(n\) is a pair \((X, {B})\), where \(X\) is a set of \(n\) vertices and \({B}\) is a collection of edge-disjoint copies of the simple graph \(G\), called blocks, which partitions the edge set of \(K_n\) (the undirected complete graph with \(n\) vertices) with vertex set \(X\). Let \((X, {B})\) be a \(G\)-design and \(H\) be a subgraph of \(G\). For each block \(B \in \mathcal{B}\), partition \(B\) into copies of \(H\) and \(G \setminus H\) and place the copy of \(H\) in \({B}(H)\) and the edges belonging to the copy of \(G \setminus H\) in \({D}(G \setminus H)\). Now, if the edges belonging to \({D}(G \setminus H)\) can be arranged into a collection \({D}_H\) of copies of \(H\), then \((X, {B}(H) \cup {D}(H))\) is a \(\lambda\)-fold \(H\)-design of order \(n\) and is called a metamorphosis of the \(\lambda\)-fold \(G\)-design \((X, {B})\) into a \(\lambda\)-fold \(H\)-design, denoted by \((G > H) – M_\lambda(n)\).
In this paper, the existence of a \((G > H) – M_\lambda(n)\) for graph designs will be presented, variations of this problem will be explained, and recent developments will be surveyed.

For an integer \(k \geq 1\) and a graph \(G = (V, E)\), a subset \(S\) of the vertex set \(V\) is \(k\)-independent in \(G\) if the maximum degree of the subgraph induced by the vertices of \(S\) is less than or equal to \(k – 1\). The \(k\)-independence number \(\beta_k(G)\) of \(G\) is the maximum cardinality of a \(k\)-independent set of \(G\). A set \(S\) of \(V\) is \(k\)-Co-independent in \(G\) if \(S\) is \(k\)-independent in the complement of \(G\). The \(k\)-Co-independence number \(\omega_k(G)\) of \(G\) is the maximum size of a \(k\)-Co-independent set in \(G\). The sequences \((\beta_k)\) and \((\omega_k)\) are weakly increasing. We define the \(k\)-chromatic number or \(k\)-independence partition number \(\chi_k(G)\) of \(G\) as the smallest integer \(m\) such that \(G\) admits a partition of its vertices into \(m\) \(k\)-independent sets and the \(k\)-Co-independence partition number \(\theta_k(G)\) of \(G\) as the smallest integer \(m\) such that \(G\) admits a partition of its vertices into \(m\) \(k\)-Co-independent sets. The sequences \((\chi_k)\) and \((\theta_k)\) are weakly decreasing. In this paper, we mainly present bounds on these four parameters, some of which are extensions of well-known classical results.

It is proved that if \(G\) is a plane embedding of a \(K_4\)-minor-free graph, then \(G\) is coupled \(5\)-choosable; that is, if every vertex and every face of \(G\) is given a list of \(5\) colours, then each of these ele-ments can be given a colour from its list such that no two adjacent or incident elements are given the same colour. Using this result it is proved also that if \(G\) is a plane embedding of a \(K_{2,3}\),\(3\)-minor-free graph or a \((\bar{K}_2 + (K_1 \cup K_2))\)-minor-free graph, then \(G\) is coupled \(5\)-choosable. All results here are sharp, even for outerplane graphs.

A Steiner system \(S(2, k, v)\) is a collection of \(k\)-subsets (blocks) of a \(k\)-set \(V\) such that each \(2\)-subset of \(V\) is contained in exactly one block. We find re-currence relations for \(S(2, k, v)\).

Denote by \(\mathcal{P}(n_1, n_2, n_3)\) the set of all polyphenyl spiders with three legs of lengths \(n_1\), \(n_2\), and \(n_3\). Let \(S^j(n_1, n_2, n_3) \in \mathcal{P}(n_1, n_2, n_3)\) (\(j \in \{1, 2, 3\}\)) be three non-isomorphic polyphenyl spiders with three legs of lengths \(n_1\), \(n_2\), and \(n_3\), and let \(m_k(G)\) and \(i_k(G)\) be the numbers of \(k\)-matchings and \(k\)-independent sets of a graph \(G\), respectively. In this paper, we show that for any \(S^j(n_1, n_2, n_3) \in \mathcal{P}(n_1, n_2, n_3)\) (\(j \in \{1, 2, 3\}\)), we have \(m_k(S_M^3(n_1, n_2, n_3)) \leq m_k(S^j(n_1, n_2, n_3)) \leq m_k(S^j(n_1, n_2, n_3))\) and \(i_k(S_O^1(n_1, n_2, n_3)) \leq i_k(S^j(n_1, n_2, n_3)) \leq i_k(S^3_M(n_1, n_2, n_3))\), with equalities if and only if \(S^j(n_1, n_2, n_3) = S_M^3(n_1, n_2, n_3)\) or \(S^j(n_1, n_2, n_3) = S_O^1(n_1, n_2, n_3)\), where \(S_O^1(n_1, n_2, n_3)\) and \(S_M^3(n_1, n_2, n_3)\) are respectively an ortho-polyphenyl spider and a meta-polyphenyl spider.