Let Pt and C` denote a path on t vertices and a cycle on ` vertices, respectively. In this paper we study the k-coloring problem for (Pt, C`)-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of P5-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that k-colorability of...

Given a graph G = (V,E) with strictly positive integer weights ωi on the vertices i ∈ V , a k-interval coloring of G is a function I that assigns an interval I(i) ⊆ {1, · · · , k} of ωi consecutive integers (called colors) to each vertex i ∈ V . If two adjacent vertices x and y have common colors, i.e. I(i)∩ I(j) 6= ∅ for an edge [i, j] in G, then the edge [i, j] is said conflicting. A k-interv...

This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k; d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to at most d-self-colored neighbors. First, (2; d) coloring is shown NP-complete for d 1, even for planar graphs, and (3; 1) coloring is also shown NP-complete for planar graphs (while there exists a...

An L(2, 1)-coloring of a graph G is a coloring of G’s vertices with integers in {0, 1, . . . , k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2, 1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of...

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k−colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for sever...

A proper k-coloring of a graph G = (V,E) is a function c : V (G) → {1, . . . , k} such that c(u) 6= c(v), for every uv ∈ E(G). The chromatic number χ(G) is the minimum k such that there exists a proper k-coloring of G. Given a spanning subgraph H of G, a q-backbone k-coloring of (G,H) is a proper k-coloring c of V (G) such that |c(u) − c(v)| ≥ q, for every edge uv ∈ E(H). The q-backbone chromat...

A hypergraph is said to be χ-colorable if its vertices can be colored with χ colors so that no hyperedge is monochromatic. 2-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 2-colorable k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer than a fraction 2−k+1 of hypered...

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most dn(G )/ke vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove tha...

For a graph G and a vertex-coloring c : V (G) → {1, 2, . . . , k}, the color code of a vertex v is the (k + 1)-tuple (a0, a1, . . . , ak), where a0 = c(v), and for 1 ≤ i ≤ k, ai is the number of neighbors of v colored i. A recognizable coloring is a coloring such that distinct vertices have distinct color codes. The recognition number of a graph is the minimum k for which G has a recognizable k...

For integers k, r > 0, a (k, r)-coloring of a graph G is a proper k-coloring c such that for any vertex v with degree d(v), v is adjacent to at least min{d(v), r} different colors. Such coloring is also called as an r-hued coloring. The r-hued chromatic number of G, χr (G), is the least integer k such that a (k, r)-coloring of G exists. In this paper, we proved that if G is a planar graph with ...