Related papers: Five-list-coloring graphs on surfaces II. A linear…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ without bichromatic paths or cycles of length four. The it star chromatic index, $\chi_{st}^{'} (G ),$ of $G$ is the minimum number $k$ for which $G$ has a star edge…

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…

An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star…

We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and…

A graph $G$ is $k$-critical if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable. A graph $G$ is $k$-choosable if $G$ has an $L$-coloring from every list assignment $L$ with $|L(v)|=k$ for all $v$, and a…

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…

Let $G$ be a graph on $n$ vertices and let $\mathcal{L}_k$ be an arbitrary function that assigns each vertex in $G$ a list of $k$ colours. Then $G$ is $\mathcal{L}_k$-list colourable if there exists a proper colouring of the vertices of $G$…

A clique-coloring of a graph $G$ is a coloring of the vertices of $G$ so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, $\mathcal{H}(G)$, of a graph $G$ has $V(G)$ as its set of vertices and the maximal…

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$…

An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where…

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number…

Let $\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a…

A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$…

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a cyclic interval $t$-coloring if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…