Related papers: On the edge chromatic vertex stability number of g…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that no path or cycle of length four is bi-colored. The star chromatic index of $G$, denoted by $\chi^{\prime}_{s}(G)$, is the minimum $k$ such that $G$ admits a star…

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring…

A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $\chi'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P.…

A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of…

The chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors. The strong chromatic index $\chi'_s(G)$ of $G$ is the smallest $k$ such that…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $\chi_{s}'(G)$ which is the minimum number of colors that allow a…

Several games that arise from graph coloring have been introduced and studied. Let $\varphi$ denote a graph invariant that arises from such a game. If $G$ is a graph and $\varphi(G-x)\neq \varphi(G)=k$, $k \geq 1$, holds true for every…

A graph \( G \) is said to be (vertex) non-repetitively colored if no simple path in \( G \) has a sequence of vertex colors that forms a repetition. Formally, a coloring \( c: V(G) \to \{1, 2, \dots, k\} \) is non-repetitive if, for every…

A proper coloring $c$ of a simple graph $G$ is harmonious if, for every pair of distinct edges $uv,xy\in E(G)$, we have that $\{c(u),c(v)\}\neq \{c(x),c(y)\}$. The harmonious chromatic number of $G$, denoted by $h(G)$, is the least positive…

We characterize stability of graph C*-algebras by giving five conditions equivalent to their stability. We also show that if G is a graph with no sources, then C*(G) is stable if and only if each vertex in G can be reached by an infinite…

A dominating set of a graph $G$ is a set of vertices $D$ such that for all $v \in V(G)$, either $v \in D$ or $(v,d) \in E(G)$ for some $d \in D$. The cardinality redundance of a vertex set $S$, $CR(S)$, is the number of vertices in $V(G)$…

Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $\chi'(G)$, and the maximum…

The \textit{packing chromatic number} of a graph $G$, denoted by $% \chi_\rho(G)$, is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In…

For an oriented graph $G$, the least number of colours required to oriented colour $G$ is called the oriented chromatic number of $G$ and denoted $\chi_o(G)$.For a non-negative integer $g$ let $\chi_o(g)$ be the least integer such that…

Given graphs $H$ and $F$, $\mathrm{ex}(n,H,F)$ denotes the largest number of copies of $H$ in $F$-free $n$-vertex graphs. Let $\chi(H)<\chi(F)=r+1$. We say that $H$ is $F$-Tur\'an-stable if the following holds. For any $\varepsilon>0$ there…

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The…

We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(\omega)$ and thus has…

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open…

An $r$-hued coloring of a simple graph $G$ is a proper coloring of its vertices such that every vertex $v$ is adjacent to at least $\min\{r, \deg(v)\}$ differently colored vertices. The minimum number of colors needed for an $r$-hued…