Related papers: Interval Total Colorings of Complete Multipartite …

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of…

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a special spanning tree in which the distance between any two adjacent vertices…

A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup…

An interval edge t-coloring of a graph G is a proper edge coloring of G with colors 1,2...,t such that at least one edge of G is colored by color i,i=1,2...,t, and the edges incident with each vertex x are colored by d_{G}(x) consecutive…

A $k$-improper edge coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that at most $k$ edges of $G$ with a common endpoint have the same color. An improper edge coloring of a graph $G$ is called an improper…

For a graph $G$, we call an edge coloring of $G$ an \textit{improper} \textit{interval edge coloring} if for every $v\in V(G)$ the colors, which are integers, of the edges incident with $v$ form an integral interval. The \textit{interval…

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…

A total coloring of a simple undirected graph $G$ is an assignment of colors to its vertices and edges such that the colors given to the vertices form a proper vertex coloring, the colors given to the edges form a proper edge coloring, and…

An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\colon E\to \mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\mathbb{Z}$. The interval thickness $\theta(G)$ of a graph…

A total colouring of a graph is a colouring of its vertices and edges such that no two adjacent vertices or edges have the same colour and moreover, no edge coloured $c$ has its endvertex coloured $c$ too. A weak total Thue colouring of a…

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be…

A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a {\it total monochromatic path} if all the edges and internal vertices on the path have the same…

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a simple connected graph $G$ is either $\Delta(G)+1$ or $\Delta(G)+2$, where $\Delta(G)$ is the largest degree of any vertex…

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer…

Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$,…

\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 $\mathcal{K}$…

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a cyclic interval $t$-coloring if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…