Related papers: A Deletion-Contraction Relation for the DP Color F…
The notion of $S$-labeling of graphs, where $S$ is a subset of a symmetric group, was introduced in 2019 by Jin, Wong, and Zhu. This notion provides the framework for a common generalization of various well studied notions of graph…
Chromatic-choosablility is a notion of fundamental importance in list coloring. A graph is chromatic-choosable when its chromatic number is equal to its list chromatic number. In 1990, Kostochka and Sidorenko introduced the list color…
In order to solve a question on list coloring of planar graphs, Dvo\v{r}\'{a}k and Postle introduced the concept of so called DP-coloring, thereby extending the concept of list-coloring. DP-coloring was anaylized in detail by Bernshteyn,…
A domination coloring of a graph $G$ is a proper vertex coloring of $G$ such that each vertex of $G$ dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all…
The online list coloring game is a two-player graph-coloring game played on a graph $G$ as follows. On each turn, a Lister reveals a new color $c$ at some subset $S \subseteq V(G)$ of uncolored vertices, and then a Painter chooses an…
A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvo\v{r}\'{a}k and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only…
Chromatic choosability is a notion of fundamental importance in list coloring. A graph $G$ is chromatic-choosable when its chromatic number, $\chi(G)$, is equal to its list chromatic number $\chi_{\ell}(G)$. In 1990, Kostochka and Sidorenko…
In this work, we introduce DPG-coloring using the concepts of DP-coloring and variable degeneracy to modify the proofs on the following papers: (i) DP-3-coloring of planar graphs without $4$, $9$-cycles and cycles of two lengths from $\{6,…
A $b$-coloring of a graph $G$ is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph $G$ has a…
We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated…
We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such…
We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least $\alpha \Delta$, where $\alpha$ is an…
The main result of this paper is the introduction of marked graphs and the marked graph polynomials ($M$-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a…
A class domination coloring (also called cd-Coloring or dominated coloring) of a graph is a proper coloring in which every color class is contained in the neighbourhood of some vertex. The minimum number of colors required for any…
A {\em balanced coloring} of a graph $G$ means a triple $\{P_1,P_2,X\}$ of mutually disjoint subsets of the vertex-set $V(G)$ such that $V(G)=P_1 \uplus P_2 \uplus X$ and $|P_1|=|P_2|$. A {\em balanced decomposition} associated with the…
Correspondence coloring, or DP-coloring, is a generalization of list coloring introduced recently by Dvo\v{r}\'{a}k and Postle. In this paper we establish a version of Dirac's theorem on the minimum number of edges in critical graphs in the…
Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…
We explore the interplay between algebraic combinatorics and algorithmic problems in graph theory by defining a polynomial with connections to correspondence colouring (also known as DP-colouring), a recent generalization of list-colouring,…
In 1980, Albertson and Berman introduced partial coloring. In 2000, Albertson, Grossman, and Haas introduced partial list coloring. Here, we initiate the study of partial coloring for an insightful generalization of list coloring introduced…
A \emph{request} on a graph assigns a preferred color to a subset of the vertices. A graph $G$ is \emph{$\epsilon$-flexibly $k$-choosable} if for every $k$-list assignment $L$ and every request $r$ on $G$, there is an $L$-coloring such that…