Related papers: An improvement on Brooks' Theorem

Let $G$ be a simple graph. Denote by $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and {\it critical}…

Let $G$ be a simple graph with maximum degree $\Delta$. A classic result of Vizing shows that $\chi'(G)$, the chromatic index of $G$, is either $\Delta$ or $\Delta+1$. We say $G$ is of \emph{Class 1} if $\chi'(G)=\Delta$, and is of…

Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined…

Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) <…

The \emph{total graph} $T(G)$ of a multigraph $G$ has as its vertices the set of edges and vertices of $G$ and has an edge between two vertices if their corresponding elements are either adjacent or incident in $G$. We show that if $G$ has…

We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $\Delta(G)\geq k + \sqrt{k}$ satisfies $\chi'(G)=\Delta(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with…

The greedy coloring algorithm shows that a graph of maximum degree at most $\Delta$ has chromatic number at most $\Delta + 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs…

Problem of finding an optimal upper bound for the chromatic no. of a (3 Times K1)-free graph is still open and pretty hard. Here we prove that for a (3 Times K1)-free graph G with maximum degree greater than or equal to 8, {\chi} is less…

In 1998 the second author proved that there is an $\epsilon>0$ such that every graph satisfies $\chi \leq \lceil (1-\epsilon)(\Delta+1)+\epsilon\omega\rceil$. The first author recently proved that any graph satisfying $\omega > \frac…

Let $\Delta$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma_2$ of $G$ with boundary $B$. Using metrical deformation via…

We prove that $K_{\chi(G)}$ is the only critical graph $G$ with $\chi(G) \geq \Delta(G) \geq 6$ and $\omega(\mathcal{H}(G)) \leq \left \lfloor \frac{\Delta(G)}{2} \right \rfloor - 2$. Here $\mathcal{H}(G)$ is the subgraph of $G$ induced on…

For a vertex subset $X$ of a graph $G$, let $\Delta_{t}(X)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k$ be a positive integer, and let $G$ be an…

Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$.…

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…

Given a graphic degree sequence $D$, let $\chi(D)$ (respectively $\omega(D)$, $h(D)$, and $H(D)$) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique…

In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with $2\leq \delta(G)\leq \Delta(G)\leq 3$ contains a maximum matching whose unsaturated vertices do not have a common neighbor, where $\Delta(G)$ and $\delta(G)$…

For a graph $G$, its energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number $\mu(G)$ is the number of edges in a maximum matching of $G$, while $\Delta$ is the maximum vertex…

The least $k$ admitting a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every $uv\in E$ is denoted by $\chi'_{\Sigma}(G)$. It has been…

A simple graph $G$ is \emph{overfull} if $|E(G)|>\Delta\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $\chi'(G)>\Delta$. The \emph{core} of a graph, denoted $G_\Delta$, is the subgraph induced by its…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…