Related papers: List chromatic numbers and singular compactness
By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]=\{1,2,\ldots, n\}$ for some integers $n\ge k\ge 1$, and in which two such sets are adjacent if and only if they realise a certain order type…
We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs. Soon after the first version was submitted to arxiv, I found out…
Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2{\omega} -…
We prove that every triangle-free graph with maximum degree $\Delta$ has list chromatic number at most $(1+o(1))\frac{\Delta}{\ln \Delta}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that…
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing…
We exhibit infinite families of planar graphs with real chromatic roots arbitrarily close to 4, thus resolving a long-standing conjecture in the affirmative.
Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable…
Recently, Balogh, Kostochka and Liu in [Packing chromatic number of cubic graphs, Discrete Math.~341 (2018) 474--483] answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded…
The clustered chromatic number of a graph class is the minimum integer $t$ such that for some $C$ the vertices of every graph in the class can be colored in $t$ colors so that every monochromatic component has size at most $C$. We show that…
We study graphs whose chromatic number is close to the order of the graph (the number of vertices). Both when the chromatic number is a constant multiple of the order and when the difference of the chromatic number and the order is a small…
The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We determine the clustered…
In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower…
We prove that any class of graphs with linear neighborhood complexity has bounded improper odd chromatic number. As a result, if $\mathcal{G}$ is the class of all circle graphs, or if $\mathcal{G}$ is any class with bounded twin-width,…
The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $\kappa$ is supercompact and the cardinals $\theta <…
A graph is weakly perfect if its clique number and chromatic number are equal. We show that the enhanced power graph of a finite group $G$ is weakly perfect: its clique number and chromatic number are equal to the maximum order of an…
In this article we consider a problem related to two famous combinatorial topics. One of them concerns the chromatic number of the space. The other deals with graphs having big girth (the length of the shortest cycle) and large chromatic…
A guaranteed upper bound is proved for the time complexity of the list-coloring problem on graphs.
Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…
A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…
Although it has recently been proved that the packing chromatic number is unbounded on the class of subcubic graphs, there exists subclasses in which the packing chromatic number is finite (and small). These subclasses include subcubic…