Related papers: Graphs, Disjoint Matchings and Some Inequalities

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…

For a graph G, let h(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected nonempty subgraphs, and let s(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected subgraphs of…

For a graph $G$ let $L(G)$ and $l(G)$ denote the size of the largest and smallest maximum matching of a graph obtained from $G$ by removing a maximum matching of $G$. We show that $L(G)\leq 2l(G),$ and $L(G)\leq (3/2)l(G)$ provided that $G$…

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…

For a graph $G$, we denote by $\sigma_{2}(G)$ the minimum degree sum of two non-adjacent vertices if $G$ is non-complete; otherwise, $\sigma_{2}(G) = +\infty$. In this paper, we prove the following two results: (i) If $s_{1}, s_{2} \ge 2$…

In a graph $G$ of maximum degree $\Delta$ let $\gamma$ denote the largest fraction of edges that can be $\Delta$ edge-coloured. Albertson and Haas showed that $\gamma \geq 13/15$ when $G$ is cubic . We show here that this result can be…

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m, F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of…

For a graph $G$ with $m$ edges, let $\rho(G)$ be its spectral radius, and let $N_F(G)$ denote the number of copies of $F$ in $G$. Nikiforov [Combin. Probab.\,Comput., 2002] proved that for $r\geq 2$, if $\rho(G)>\sqrt{(1-1/r)2m}$, then…

Let $G=(V,E)$ be a $\tau$-critical graph with $\tau(G)=t$. Erd\H{o}s and Gallai proved that $|V|\leq 2t$ and the bound $|E|\leq {t+1\choose 2}$ was obtained by Erd\H{o}s, Hajnal and Moon. We give here the sharp combined bound $|E|+|V|\leq…

A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied…

The \emph{genus} $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. A drawing of a graph on a surface is \emph{independently even} if every pair of nonadjacent…

In \cite{2012a}, Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$ as \hskip3.3cm $\rm irr_{t}$$(G) = \frac{1}{2}\sum_{u,v\in V}|d_{G}(u)-d_{G}(v)|, $ \noindent where $d_{G}(u)$ denotes the vertex degree of a vertex $u\in…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G)\geq9$, then $\chi(G)\leq\max\{\Delta(G)-1,\omega(G)\}$. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let…

Let $G=(V(G), E(G))$ be a graph with maximum degree $\Delta$. For a subset $M$ of $E(G)$, we denote by $G[V(M)]$ the subgraph of $G$ induced by the endvertices of edges in $M$. We call $M$ a semistrong matching if each edge of $M$ is…

Let $G$ be a graph with $m$ edges and let $\mathrm{mc}(G)$ denote the size of a largest cut of $G$. The difference $\mathrm{mc}(G)-m/2$ is called the surplus $\mathrm{sp}(G)$ of $G$. A fundamental problem in MaxCut is to determine…

A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal…

Here we prove that for a 2K2-free graph G with maximum degree greater than or equal to 5, the chromatic number is less than or equal to max{maximum degree-1, maximum clique size}. This implies that Borodin & Kostochka Conjecture is true for…

Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that $\textit{cr}(G)=O(\mathop{\mathrm{pcr}}(G)^{3/2})$ for every…

An \emph{equitable coloring} of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $\Delta \geq 2$ has an…