A graceful labeling of a graph \(G\) with \(m\) edges is a function \(f: V(G) \to \{0, \ldots, m\}\) such that distinct vertices receive distinct numbers and \(\{|f(u) – f(v)|: uv \in E(G)\} = \{1, \ldots, m\}\). A graph is graceful if it has a graceful labeling. In \([1]\) this question was posed: “Is there an \(n\)-chromatic graceful graph for \(n \geq 6\)?”. In this paper it is shown that for any natural number \(n\), there exists a graceful graph \(G\) with \(\chi(G) = n\).

A connected graph \(G = (V, E)\) is said to be \((a, d)\)-antimagic, for some positive integers \(a\) and \(d\), if its edges admit a labeling by all the integers in the set \(\{1, 2, \ldots, |E(G)|\}\) such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term \(a\) and the common difference \(d\). Mirka Miller and Martin Ba\'{e}a proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+3}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). The first author of this paper proved that \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 6\). In this paper, we show that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2} , 3)\)-antimagic for \(n \equiv 2 \pmod{4}\), \(n \geq 10\).

Let \(0 \leq p \leq [\frac{r+1}{2}]\) and \(\sigma(K_{r+1}^{-p},n)\) be the smallest even integer such that each \(n\)-term graphic sequence with term sum at least \(\sigma(K_{r+1}^{-p},n)\) has a realization containing \(K_{r+1}^{-p}\) as a subgraph, where \(K_{r+1}^{-p}\) is a graph obtained from a complete graph \(K_{r+1}\) on \(r+1\) vertices by deleting \(p\) edges which form a matching. In this paper, we determine \(\sigma(K_{r+1}^{-p},n)\) for \(r \geq 2, 1 \leq p \leq [\frac{r+1}{2}]\) and \(n \geq 3r + 3\). As a corollary, we also determine \(\sigma(K_{1^*,2^t}n)\) for \(t \geq 1\) and \(n \geq 3s + 6t\), where \(K_{1^*,2^t}\) is an \(r_1\times r_2\times \ldots \times r_{s+t}\) complete \((s + t)\)-partite graph with \(r_1 = r_2 = \ldots = r_s = 1\) and \(r_{s+1} = r_{s+2} = \ldots = r_{s+t} = 2\) and \(\sigma(K_{1^*,2^t},n)\) is the smallest even integer such that each \(n\)-term graphic sequence with term sum at least \(\sigma(K_{1^*,2^t},n)\) has a realization containing \(K_{1^*,2^t}\) as a subgraph.

Let \(n, s_1\) and \(s_2\) be positive integers such that \(1 \leq s_1 \leq n/2, 1 \leq s_2 \leq n/2, s_1 \neq s_2\) and \(gcd(n, s_1, s_2) = 1\). An undirected double-loop network \(G(n;\pm s_1,\pm s_2)\) is a graph \((V, E)\), where \(V = \mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\), and \(E = \{(i \to i+s_1 \mod n), (i\to i-s_1 \mod n), (i\to i+s_2 \mod n), (i\to i-s_2 \mod n) | i = 0, 1, 2, \ldots, n-1\}\). In this paper, a diameter formula is given for an undirected double-loop network \(G(n; \pm s_1, \pm s_2)\). As its application, two new optimal families of undirected double-loop networks are presented.

Anthony J. Macula constructed a \(d\)-disjunct matrix \(\delta(n,d,k)\) in \([1]\), and we now know it is determined by one type of pooling space. In this paper, we give some properties of \(\delta(n,d,k)\) and its complement \(\delta^c(n,d,k)\).

Let \(S\) be a primitive non-powerful signed digraph. The base \(l(S)\) of \(S\) is the smallest positive integer \(l\) such that for all ordered pairs of vertices \(i\) and \(j\) (not necessarily distinct), there exists a pair of \(SSSD\) walks of length \(t\) from \(i\) to \(j\) for each integer \(t \geq l\). In this work, we use \(PNSSD\) to denote the class of all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop. Let \(l(n)\) be the largest value of \(l(S)\) for \(S \in\) \(PNSSD\), and \(L(n) = \{l(S) | S \in PNSSD\}\). For \(n \geq 3\), we show \(L(n) = \{2, 3, \ldots, 2n\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop whose bases attain \(l(n)\).

For a graph \(G = (V, E)\) and a binary labeling \(f : V(G) \to \mathbb{Z}_2\), let \(v_f(i) = |f^{-1}(1)|\). The labeling \(f\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). Any vertex labeling \(f : V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f^* : E(G) \to \mathbb{Z}_2\) defined by \(f^*(xy) =| f(x) – f(y)|\). Let \(e_f(i) = |f^{*-1}(i)|\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined by

\[FI(G) = \{|e_f(1) – e_f(0)| : f \text{ is a friendly vertex labeling of } G\}.\]

In \([15]\) Lee and Ng conjectured that the friendly index sets of trees will form an arithmetic progression. This conjecture has been mentioned in \([17]\) and other manuscripts. In this paper, we will first determine the friendly index sets of certain caterpillars of diameter four. Then we will disprove the conjecture by presenting an infinite number of trees whose friendly index sets do not form an arithmetic progression.

Let \(S\) be a primitive non-powerful signed digraph of order \(n\). The base of a vertex \(u\), denoted by \(l_S(u)\), is the smallest positive integer \(l\) such that there is a pair of SSSD walks of length \(i\) from \(u\) to each vertex \(v \in V(S)\) for any integer \(t \geq l\). We choose to order the vertices of \(S\) in such a way that \(l_S(1) \leq l_S(2) \leq \ldots \leq l_S(n)\), and call \(l_S(k)\) the \(k\)th local base of \(S\) for \(1 \leq k \leq n\). In this work, we use PNSSD to denote the class of all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop. Let \(l(k)\) be the largest value of \(l_S(k)\) for \(S \in\) PNSSD, and \(L(k) = \{l_S(k) | S \in PNSSD\}\). For \(n \geq 3\) and \(1 \leq k \leq n-1\), we show \(I(k) = 2n – 1\) and \(L(k) = \{2, 3, \ldots, 2n-1\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs whose \(k\)th local bases attain \(I(k)\).

Let \(\mathcal{U}_n(k)\) denote the set of all unicyclic graphs on \(n\) vertices with \(k\) (\(k \geq 1\)) pendant vertices. Let \(\diamondsuit_4^k\) be the graph on \(n\) vertices obtained from \(C_4\) by attaching \(k\) paths of almost equal lengths at the same vertex. In this paper, we prove that \(\diamondsuit_4^k\) is the unique graph with the largest Laplacian spectral radius among all the graphs in \(\mathcal{U}_n(k)\), when \(n \geq k + 4\).

For graphs \(G_1, G_2, \ldots, G_m\), the Ramsey number \(R(G_1, G_2, \ldots, G_m)\) is defined to be the smallest integer \(n\) such that any \(m\)-coloring of the edges of the complete graph \(K_n\) must include a monochromatic \(G_i\) in color \(i\), for some \(i\). In this note, we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral \(C_4\), including:\(R(C_4, K_9) \leq 32,
19 \leq R(C_4, C_4, K_4)\leq 22, 31 \leq R(C_4, C_4, C_4, K_4) \leq 50, 52 \leq R(C_4, K_4, K_4) \leq 72, 42 \leq R(C_4, C_4, K_3, K_5) \leq 76, 87\leq R(C_4, C_4, K_4, K_4) \leq 179.\)