Ewa Wojcicka (Journal of Graph Theory, \(14(1990), 205-215)\) showed that every connected, 3-color-critical graph on more than 6 vertices has a Hamiltonian path. Henning et al. (Discrete Mathematics, \(161(1996), 175-184)\) defined a graph \(G\) to be \(k\)-\((\gamma, d)\)-critical graph if \(\gamma(G) = k\) and \(\gamma(G + uv) = k – 1\) for each pair \(u, v\) of nonadjacent vertices of \(G\) that are at distance at most \(d\) apart. They asked if a 3-\((\gamma, 2)\)-critical graph must contain a dominating path. In this paper, we show that every connected, 3-\((\gamma, 2)\)-critical graph must contain a dominating path. Further, we show that every connected, 3-\((\gamma, 2)\)-critical graph on more than 6 vertices has a Hamiltonian path.

Let \(d(u,v)\) denote the distance between two distinct vertices of a connected graph \(G\) and \(diam(G)\) be the diameter of \(G\). A radio labeling \(f\) of \(G\) is an assignment of positive integers to the vertices of \(G\) satisfying \(d(u,v) + |f(u) – f(v)| \geq diam(G) + 1\). The maximum integer in the range of the labeling is its span. The radio number of \(G\), denoted by \(rn(G)\), is the minimum possible span. In \([7]\) M. Farooq et al. found the lower bound for the radio number of generalized gear graph. In this paper, we give an upper bound for the radio number of generalized gear graph, which coincides with the lower bound found in \([7]\).

In this paper, we study codes over polynomial rings and establish a connection to Jacobi Hilbert modular forms, specifically Hilbert modular forms over the totally real field via the complete weight enumerators of codes over polynomial rings.

In this paper, we introduce the notion of \((\alpha, \beta)\)-generalized \(d\)-derivations on lattices and investigate some related properties. Also, using the notion of permuting \((\alpha, \beta)\)-triderivation, we characterize the distributive elements of a lattice.

Suppose \(\{P_r\}\) is a nonempty family of paths for \(r \geq 3\), where \(P_r\) is a path on \(r\) vertices. An \(r\)-coloring of a graph \(G\) is said to be \(\{P_r\}\)-free if \(G\) contains no 2-colored subgraph isomorphic to any path \(P_r\) in \(\{P_r\}\). The minimum \(k\) such that \(G\) has a \(\{P_r\}\)-free coloring using \(k\) colors is called the \(\{P_r\}\)-free chromatic number of \(G\) and is denoted by \(\chi_{\{P_r\}}(G)\). If the family \(\{P_r\}\) consists of a single graph \(P_r\), then we use \(\chi_{P_r}(G)\). In this paper, \(\{P_r\}\)-free colorings of Sierpiński-like graphs are considered. In particular, \(\chi_{P_3}(S_n)\), \(\chi_{P_4}(S_n)\), \(\chi_{P_4}(S(n, k))\), \(\chi_{P_3}(S^{++}(n, k))\), and \(\chi_{P_4}(S^{++}(n, k))\) are determined.

Let \(G = (V,E)\) be a graph with \(v = |V(G)|\) vertices and \(e = |E(G)|\) edges. An \((a, d)\)-edge-antimagic total labeling of the graph \(G\) is a one-to-one map \(A\) from \(V(G) \cup E(G)\) onto the integers \(\{1,2,\ldots,v+e\}\) such that the set of edge weights of the graph \(G\), \(W = \{w(xy) : xy \in E(G)\}\) form an arithmetic progression with the initial term \(a\) and common difference \(d\), where \(w(xy) =\lambda(x) + \lambda(y) + \lambda(xy)\) for any \(xy \in E(G)\). If \(\lambda(V(G)) = \{1,2,\ldots,v\}\) then \(G\) is super \((a, d)\)-edge-antimagic total, i.e., \((a,d)\)-EAT. In this paper, for different values of \(d\), we formulate super \((a, d)\)-edge-antimagic total labeling on subdivision of stars \(K_{1,p}\) for \(p \geq 5\).

We discuss the chromaticity of one family of \(K_4\)-homeomorphs which has girth \(7\) and has exactly \(1\) path of length \(1\), and give a sufficient and necessary condition for the graphs in the family to be chromatically unique.

A theta graph is denoted by \(\theta(a,b,c)\), where \(a \leq b \leq c\). It is obtained by subdividing the edges of the multigraph consisting of \(3\) parallel edges \(a\) times, \(b\) times, and \(c\) times each. In this paper, we show that the theta graph is matching unique when \(a \geq 2\) or \(a = 0\), and all theta graphs are matching equivalent when only one of the edges is subdivided one time. We also completely characterize the relation between the largest matching root \(\alpha\) and the length of path \(a, b, c\) of a theta graph, and determine the extremal theta graphs.

The line graph of \(G\), denoted \(L(G)\), is the graph with vertex set \(E(G)\), where vertices \(x\) and \(y\) are adjacent in \(L(G)\) if and only if edges \(x\) and \(y\) share a common vertex in \(G\). In this paper, we determine all graphs \(G\) for which \(L(G)\) is a circulant graph. We will prove that if \(L(G)\) is a circulant, then \(G\) must be one of three graphs: the complete graph \(K_4\), the cycle \(C_n\), or the complete bipartite graph \(K_{a,b}\), for some \(a\) and \(b\) with \(\gcd(a,b) = 1\).