相关论文: A counterexample to a conjecture of Bj\"{o}rner an…

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich,…

In the vertex colouring game on a graph $G$, Maker and Breaker alternately colour vertices of $G$ from a palette of $k$ colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker…

Given a graph $G =(V,E)$, a bijection $f: E \rightarrow \{1, 2, \dots,|E|\}$ is called a local antimagic labeling of $G$ if the vertex weight $w(u) = \sum_{uv \in E} f(uv)$ is distinct for all adjacent vertices. The vertex weights under the…

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…

Given an arbitrary graph $G$ we study the chromatic number of a random subgraph $G_{1/2}$ obtained from $G$ by removing each edge independently with probability $1/2$. Studying $\chi(G_{1/2})$ has been suggested by Bukh~\cite{Bukh}, who…

A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges…

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all…

The union vertex-distinguishing chromatic index $\chi'_\cup(G)$ of a graph $G$ is the smallest natural number $k$ such that the edges of $G$ can be assigned nonempty subsets of $[k]$ so that the union of the subsets assigned to the edges…

The reconfiguration graph $\mathcal{C}_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $\mathcal{C}_k(G)$ are adjacent precisely when those $k$-colourings differ on a single…

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'{a}k and Postle. In this paper we introduce and study the fractional DP-chromatic number $\chi_{DP}^\ast(G)$. We…

We study relations between three interrelated notions of graph (list) coloring: single conflict coloring, adapted list coloring and choosability with separation (with $1$ overlapping color between lists of adjacent vertices), and their…

By a $z$-coloring of a graph $G$ we mean any proper vertex coloring consisting of the color classes $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1 \leq i < j \leq k$, any vertex of color $j$ is adjacent to a…

Let $h(G)$ denote the largest $t$ such that $G$ contains $K_t$ as a minor and $\chi(G)$ be the chromatic number of $G$ respectively. In 1943, Hadwiger conjectured that $h(G) \geq \chi(G)$ for any graph $G$. In this paper, we prove that…

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 $G$.…

The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs…

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…

We prove a conjecture of Ohba which says that every graph $G$ on at most $2\chi(G)+1$ vertices satisfies $\chi_\ell(G)=\chi(G)$.

A well known observation of Lov\'asz is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This very simple criterion turned out to be extremly useful . In this short paper we…

A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of…

An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits…