Related papers: Counting subgraphs of coloring graphs
Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex…
A complete $k$-coloring of a graph $G$ is a (not necessarily proper) $k$-coloring of the vertices of $G$, such that each pair of different colors appears in an edge. A complete $k$-coloring is also called connected, if each color class…
A {\em chromatic root} is a root of the chromatic polynomial of a graph. While the real chromatic roots have been extensively studied and well understood, little is known about the {\em real parts} of chromatic roots. It is not difficult to…
Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for which a greedy coloring of every induced subgraph uses only the minimally necessary number of colors $\chi(G)$. In recent work on reciprocal best match graphs…
In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously…
Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…
Let $G$ be a simple graph. The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated…
In this paper, we study the problem of partitioning a graph into connected and colored components called blocks. Using bivariate generating functions and combinatorial techniques, we determine the expected number of blocks when the vertices…
A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL…
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…
Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be…
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a…
The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…
A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph $G$ to a 2-edge-colored graph $H$ is a mapping $\varphi: V(G) \rightarrow V(H)$ that maps every edge in $G$ to an…
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…
An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits…
The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the…
A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The…
In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…
A total $k$-coloring of a graph $G$ is a coloring of $V(G)\cup E(G)$ using $k$ colors such that no two adjacent or incident elements receive the same color. The total chromatic number $\chi"(G)$ of $G$ is the smallest integer $k$ such that…