Related papers: Choosability with Separation of Cycles and Outerpl…

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al.,…

A proper $s$-coloring of an $n$-vertex graph is \emph{equitable} if every color class has size $\lfloor{n/s}\rfloor$ or $\lceil{n/s}\rceil$. A necessary condition to have an equitable $s$-coloring is that every vertex $v$ appears in an…

We study the problem of determining whether a given graph~$G=(V,E)$ admits a matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently whether~$G-M$ is bipartite). This problem is equivalent to determine whether~$G$ admits…

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be…

An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring}…

An $L(2, 1)$-labeling of a graph $G$ is an assignment of a nonnegative integer to each vertex of $G$ such that adjacent vertices receive integers that differ by at least two and vertices at distance two receive distinct integers. The span…

Given positive integers $p \ge k$, and a non-negative integer $d$, we say a graph $G$ is $(k,d,p)$-choosable if for every list assignment $L$ with $|L(v)|\geq k$ for each $v \in V(G)$ and $|\bigcup_{v\in V(G)}L(v)| \leq p$, there exists an…

We study the hat guessing game on graphs. In this game, a player is placed on each vertex $v$ of a graph $G$ and assigned a colored hat from $h(v)$ possible colors. Each player makes a deterministic guess on their hat color based on the…

We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment…

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The…

For a graph $G$, we call an edge coloring of $G$ an \textit{improper} \textit{interval edge coloring} if for every $v\in V(G)$ the colors, which are integers, of the edges incident with $v$ form an integral interval. The \textit{interval…

The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour…

Dvo\v{r}\'ak \emph{et al.} introduced a variant of the Randi\'c index of a graph $G$, denoted by $R'(G)$, where $R'(G)=\sum_{uv\in E(G)}\frac 1 {\max\{d(u), d(v)\}}$, and $d(u)$ denotes the degree of a vertex $u$ in $G$. The coloring number…

A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…

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 {\em conflict-free coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed)…

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that…

Given a graph $G(V,E)$, a vertex subset $S$ of $G$ is called an open packing in $G$ if no pair of distinct vertices in $S$ have a common neighbour in $G$. The size of a largest open packing in $G$ is called the open packing number,…

Let $G=(V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$.…

Let $G$ be a graph with $m$ edges and let $\mathrm{mc}(G)$ denote the size of a largest cut of $G$. The difference $\mathrm{mc}(G)-m/2$ is called the surplus $\mathrm{sp}(G)$ of $G$. A fundamental problem in MaxCut is to determine…