Related papers: A sharp upper bound for the harmonious total chrom…

Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $\delta$, and maximum degree $\Delta$. The "oriented chromatic number" of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $\chi_{\rm pcf}^h(G)$ denote the minimum $k$ such that…

A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper conflict-free…

Total coloring of a graph is a coloring of its vertices and edges such that adjacent or incident elements receive distinct colors. Total coloring conjecture (stipulating that the total chromatic number of a graph $G$ is at most…

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph…

The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $\varphi:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one…

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…

Let $pr(K_{n}, G)$ be the maximum number of colors in an edge-coloring of $K_{n}$ with no properly colored copy of $G$. In this paper, we show that $pr(K_{n}, G)-ex(n, \mathcal{G'})=o(n^{2}), $ where $\mathcal{G'}=\{G-M: M \text{ is a…

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…

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…

For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We…

Let $G$ be a graph of maximum degree $\Delta$ which does not contain isolated vertices. An edge coloring $c$ of $G$ is called conflict-free if each edge's closed neighborhood includes a uniquely colored element. The least number of colors…

This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable…

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…

Let $G$ be an edge-coloured graph. The minimum colour degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly…

\textit{Total Coloring} of a graph is a major coloring problem in combinatorial mathematics, introduced in the early $1960$s. A \textit{total coloring} of a graph $G$ is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$…

Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A…