A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k ≥ 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we ...

Let G be an edge-colored graph. A tree T in G is a proper tree if no two adjacent edges of it are assigned the same color. Let k be a fixed integer with 2 ≤ k ≤ n. For a vertex subset S ⊆ V (G) with |S| ≥ 2, a tree is called an S-tree if it connects the vertices of S in G. A k-proper coloring of G is an edge-coloring of G having the property that for every set S of k vertices of G, there exists...

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension on diameter at most $d$, proving $\textsf{NP}$-completeness in cases, leaving open only List $3$-Coloring $3$-Precoloring when $d=3$.
 Some these results are obtained $\textsc{through}$ proof that Surjective $C_6$-Ho...

Many heuristic algorithms have been proposed for graph coloring. The simplest is perhaps the greedy algorithm. Many variations have been proposed for this algorithm at various levels of sophistication, but it is generally assumed that the coloring will occur in a single attempt. We note that if a new permutation of the vertices is chosen which respects the independent sets of a previous colorin...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is sdegenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerate graph G has a total coloring with ∆+1 colors if the maximum degre...

Given an undirected graph G = (V, E), the Graph Coloring Problem (GCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. A subset of V is called a stable set if no two adjacent vertices belong to it. A k coloring of G is a coloring which uses k colors, and corresponds to a partition of V into k stab...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Many combinatorial problems can be efficiently solved for partial k-trees (graphs of treewidth bounded by a constant k). However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a giv...

In a recent article [5], the authors claim that the distance between the b-chromatic index of a tree and a known upper bound is at most 1. At the same time, in [7] the authors claim to be able to construct a tree where this difference is bigger than 1. However, the given example was disconnected, i.e., actually consisted of a forest. Here, we slightly modify their construction in order to produ...

A k-list assignment L of a graph G is a mapping which assigns to each vertex v of G a set L(v) of size k. A (k,t)-list assignment of G is a k-list assignment with | ⋃ v∈V (G) L(v)| = t. An L-coloring φ of G is a proper coloring of G such that φ(v) is chosen from L(v) for every vertex v. A graph G is Lcolorable if G has an L-coloring. When the parameter t is not of special interest, we simply sa...