Related papers: A $4$-choosable graph that is not $(8:2)$-choosabl…

We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. B\"achle, A.…

A graph $G$ is said to be $k$-critical if $G$ is $k$-colorable and $G-e$ is not $k$-colorable for every edge $e$ of $G$. In this paper, we present some new methods from two or more small 4-critical graphs to construct a larger 4-critical…

Recently, Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$th power graphs are chromatic-choosable. We answer this question in the…

A graph $G$ is free $(a,b)$-choosable if for any vertex $v$ with $b$ colors assigned and for any list of colors of size $a$ associated with each vertex $u\ne v$, the coloring can be completed by choosing for $u$ a subset of $b$ colors such…

Twin-width is a recently introduced graph parameter for finite graphs. It is an open problem to determine whether there is an $n$-vertex graph having twin-width at least $n/2$ (due to J. Ahn, K. Hendrey, D. Kim and S. Oum). In an earlier…

For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges…

In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. After a series of papers confirming the conjecture for a number of graph classes, the conjecture is yet refuted by Lafond and Seamone in 2015. Given that…

We say that a graph $H$ is planar unavoidable if there is a planar graph $G$ such that any red/blue coloring of the edges of $G$ contains a monochromatic copy of $H$, otherwise we say that $H$ is planar avoidable. I.e., $H$ is planar…

A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. A graph is equitably $k$-colorable if the vertex set…

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels…

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least…

In the List $k$-Coloring problem we are given a graph whose every vertex is equipped with a list, which is a subset of $\{1,\ldots,k\}$. We need to decide if $G$ admits a proper coloring, where every vertex receives a color from its list.…

A mixed graph is, informally, an object obtained from a simple undirected graph by choosing an orientation for a subset of its edges. A mixed graph is $(m, n)$-coloured if each edge is assigned one of $m \geq 0$ colours, and each arc is…

We give results concerning two problems on the recently introduced \textit{flip colourings of graphs}. For positive integers $b, r$ with $b < r$, we say that a $b + r$ regular graph is a $(b,r)$-\textit{flip graph} if there exists a…

Hu and Li investigate the signed graph version of Erd$\ddot{\mathrm{o}}$s problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\leq k\leq c$, is $3$-colorable and prove that each signed planar…

Equitable $k$-choosability is a list analogue of equitable $k$-coloring that was introduced by Kostochka, Pelsmajer, and West in 2003. It is known that if vertex disjoint graphs $G_1$ and $G_2$ are equitably $k$-choosable, the disjoint…

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper…

Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the vertex…

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of…

We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…