Related papers: Coloring graphs with forbidden bipartite subgraphs

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our…

Let $G$ be a graph of order $n$, maximum degree at most $\Delta$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function…

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…

For graphs $G$ and $H$, a homomorphism from $G$ to $H$, or $H$-coloring of $G$, is a map from the vertices of $G$ to the vertices of $H$ that preserves adjacency. When $H$ is composed of an edge with one looped endvertex, an $H$-coloring of…

A celebrated result of Johansson in graph theory states that every triangle-free graph of maximum degree $\Delta$ can be properly colored with $O(\Delta/\ln\Delta)$ colors, improving upon the "greedy bound" of $\Delta+1$ coloring in general…

A graph $G$ is $(d_1,\ldots,d_k)$-colorable if its vertex set can be partitioned into $k$ sets $V_1,\ldots,V_k$, such that for each $i\in\{1, \ldots, k\}$, the subgraph of $G$ induced by $V_i$ has maximum degree at most $d_i$. The Four…

Let $G$ be a graph, $\chi(G)$ be the minimal number of colors which can be assigned to the vertices of $G$ in such a way that every two adjacent vertices have different colors and $\omega(G)$ to be the least upper bound of the size of the…

A \emph{coloring} of a graph $G$ is a map $f:V(G)\to \mathbb{Z}^+$ such that $f(v)\ne f(w)$ for all $vw\in E(G)$. A coloring $f$ is an \emph{odd-sum} coloring if $\sum_{w\in N[v]}f(w)$ is odd, for each vertex $v\in V(G)$. The \emph{odd-sum…

Let $G$ be any triangle-free graph with maximum degree $\Delta\leq 3$. Staton proved that the independence number of $G$ is at least 5/14n. Heckman and Thomas conjectured that Staton's result can be strengthened into a bound on the…

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le \Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $\Delta$ in which the neighbourhood of every vertex in $G$ spans at most $\Delta^2/f$ edges, (i) an…

We give a uniform and self-contained proof that if $G$ is a connected graph with $\chi(G) = \Delta(G)$ and $G\neq \overline{C_7}$, then $G$ contains either $K_{\Delta(G)}$ or an odd hole where every vertex has degree at least $\Delta(G)-1$…

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. In 2019, Bernshteyn, Kostochka, and Zhu introduced a fractional version of DP-coloring. They showed that…

Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$…

A strong edge-coloring $\varphi$ of a graph $G$ assigns colors to edges of $G$ such that $\varphi(e_1)\ne \varphi(e_2)$ whenever $e_1$ and $e_2$ are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of…

We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to $t$-core (subgraph induced by the vertices $v$ with $d(v)+\mu(v)> \Delta+t$), and find a sufficient condition for $(\Delta+t)$-edge-coloring.…

Erd\H{o}s and Simonovits asked the following question: For an integer $c\geq 2$ and a family of non-bipartite graphs $\mathcal{F}$, what is the infimum of $\alpha$ such that any $\mathcal{F}$-free $n$-vertex graph with $n$ large enough and…

The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…

A simple graph $G$ with maximum degree $\Delta$ is overfull if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The core of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. Clearly, the chromatic index…

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds…

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, \Delta)$ be the largest $k$…