Let \(G\) be a disconnected tripartite unicyclic graph on seven edges with two or more connected components. We prove that \(G\) decomposes the complete graph \(K_{n}\) whenever \(n\equiv0,1\pmod{14}\) using labeling techniques.

Let \(G\) be a graph. We introduce the balanced antimagic labeling as an analogue to the antimagic labeling. A \(k\)-total balanced antimagic labelling is a map \(c\colon V (G)\cup E(G) \to \{1,2,\ldots,k\}\) such that: the label classes differ in size by at most one, each vertex \(x\) is assigned the weight \(w(x)={c}(x)+\sum\limits_{x\in e}{c}(e)\), and \(w(x)\neq w(y)\) for \(x\neq y\).

We present several properties of balanced antimagic labeling. We also derive such a labeling for complete graphs and complete bipartite graphs.

For a subgraph \(G\) of the complete graph \(K_n\), a \(G\)-design of order \(n\) is a partition of the edges of \(K_n\) into edge-disjoint copies of \(G.\) For a given graph \(G\), the \(G\)-design spectrum problem asks for which \(n\) a \(G\)-design of order \(n\) exists. This problem has recently been completely solved for every graph \(G\) with less than seven edges, with the lone exception of \(G \cong K_3 \cup 2K_2,\) the disconnected graph consisting of a triangle and two isolated edges. In this article, we solve this problem by proving that a \(K_3 \cup 2K_2\)-design of order \(n\) exists if and only if \(n \equiv 0 \; \textrm{or} \; 1 \pmod{5}\) and \(n\geq 10.\)

A question involving a chess piece called a prince on the \(8\times 8\) chessboard leads to a concept in graph theory involving total domination in the Cartesian products of paths and cycles. A vertex \(u\) in a graph \(G\) totally dominates a vertex \(v\) if \(v\) is adjacent to \(u\). A subset \(S\) of the vertex set of a graph \(G\) is a total dominating set for \(G\) if every vertex of \(G\) is totally dominated by at least one vertex of \(S\). If \(S\) is a total dominating set of a graph \(G\), then \(S(v)\) denotes the number of vertices in \(S\) that totally dominate \(v\). A total dominating set \(S\) in a graph \(G\) is called a proper total dominating set if \(S(u) \ne S(v)\) for every two adjacent vertices \(u\) and \(v\) of \(G\). It is shown that \(C_n \ \Box \ K_2\) possesses a proper total dominating set if and only if \(n\ge 4\) is even and the graph \(C_n \ \Box \ P_m\) possesses a proper total dominating set for every even integer \(n \ge 4\) and every integer \(m \ge 3\). Furthermore, \(C_3 \ \Box \ P_m\) possesses a proper total dominating set if and only if \(m = 3\). If \(n\ge 5\) is an odd integer and \(m\equiv 3 \pmod 4\), then \(C_n \ \Box \ P_m\) has a proper total dominating set. If at least one of \(n\) and \(m\) is even, then \(C_n \ \Box \ C_m\) has a proper total dominating set. The graphs \(C_n \ \Box \ C_m\) are further studied when both \(n\) and \(m\) are odd. Other results and questions are also presented.

For a graph \(F\) and a positive integer \(t\), the vertex-disjoint Ramsey number \(VR_t(F)\) is the minimum positive integer \(n\) such that every red-blue coloring of the edges of the complete graph \(K_n\) of order \(n\) results in \(t\) pairwise vertex-disjoint monochromatic copies of subgraphs isomorphic to \(F\), while the edge-disjoint Ramsey number \(ER_t(F)\) is the corresponding number for edge-disjoint subgraphs. These numbers have been investigated for the three connected graphs \(K_3\), \(P_4\) and \(K_{1, 3}\) of size 3. For two vertex-disjoint graphs \(G\) and \(H\), let \(G+H\) denote the union of \(G\) and \(H\). Here we study these numbers for the two disconnected graphs \(3K_2\) and \(P_3+P_2\) of size 3. It is shown that \(VR_t(3K_2)= 6t+2\) and \(VR_t(P_3+P_2)= 5t+1\) for every positive integer \(t\). The numbers \(ER_t(3K_2)\) and \(ER_t(P_3+P_2)\) are determined for \(t \le 4\) and bounds are established for \(ER_t(3K_2)\) and \(ER_t(P_3+P_2)\) when \(t \ge 5\). Other results and problems are presented as well.

Every red-blue coloring of the edges of a graph \(G\) results in a sequence \(G_1\), \(G_2\), \(\ldots\), \(G_{\ell}\) of pairwise edge-disjoint monochromatic subgraphs \(G_i\) (\(1 \le i \le \ell\)) of size \(i\) such that \(G_i\) is isomorphic to a subgraph of \(G_{i+1}\) for \(1 \le i \le \ell-1\). Such a sequence is called a Ramsey chain in \(G\) and \(AR_c(G)\) is the maximum length of a Ramsey chain in \(G\) with respect to a red-blue coloring \(c\). The Ramsey index \(AR(G)\) of \(G\) is the minimum value of \(AR_c(G)\) among all red-blue colorings \(c\) of \(G\). Several results giving the Ramsey indexes of graphs are surveyed. A galaxy is a graph each of whose components is a star. Results and conjectures on Ramsey indexes of galaxies are presented.

An edge-coloring of a graph \(G\) with natural numbers \(1,2,\ldots\) is called a sum edge-coloring if the colors of edges incident to any vertex of \(G\) are distinct and the sum of the colors of the edges of \(G\) is minimum. The edge-chromatic sum of a graph \(G\) is the sum of the colors of edges in a sum edge-coloring of \(G\). In general, the problem of finding the edge-chromatic sum of an \(r\)-regular (\(r\geq 3\)) graph is \(NP\)-complete. In this paper we provide some bounds on the edge-chromatic sums of various products of graphs. In particular, we give tight upper bounds on the edge-chromatic sums of tensor, strong tensor, Cartesian, strong products and composition of graphs. We also determine the edge-chromatic sums and edge-strengths of the Cartesian products of regular graphs and paths (cycles) with an even number of vertices. Finally, we determine the edge-chromatic sums and edge-strengths of grids, cylinders, and tori.

Generalised nice sets are defined as subsets of edges of the extended Fano plane satisfying a certain absorbing property. The classification up to collineations, purely combinatorial in nature, provides 245 generalised nice sets. In turn, this gives rise to new Lie algebras obtained by modifying the bracket of homogeneous elements on an initial \(\mathbb Z_2^3\)-graded Lie algebra.

Let \(\alpha(n)\) denote the number of perfect square permutations in the symmetric group \(S_n\). The conjecture \(\alpha(2n+1) = (2n+1) \alpha(2n)\), provided by Stanley [4], was proved by Blum [1] using generating functions. However, several structural questions about these special permutations remained open. This paper presents an alternative and constructive proof for this conjecture, which highlights the deeper interplay between cycle structures and square properties. At the same time, we demonstrate that all permutations with an even number of even cycles in both \(S_{2n}\) and \(S_{2n+1}\) can be categorized into three disjoint types that correspond to each other.

In this paper, we generalize the \(k\)-Jacobsthal sequences and call them the generalized \((k,t)\)-Jacobsthal \(p\)-sequences. Also, we obtain combinatorial identities. Then, the generalized\((k,t)\)-Jacobsthal \(p\)-matrix is used to factorize the Pascal matrix. Finally, using the Riordan method, we obtain two factorizations of the Pascal matrix involving the generalized \((k,t)\)-Jacobsthal \(p\)-sequences.