A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c : X → [k] such that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. Upper chromatic number is the maximum number of colors that can be used in a ...

The b-chromatic number of a graph G, χb(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the b-Chromatic Number problem, the objective is to decide whether χb(G) ≥ k. Testing whether χb(G) = ∆(G) + 1, where ∆(G) is the maximum degree of a graph, it...

The dichromatic number ~ χ(D) of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this paper, we study the li...

Hq(n, d) is defined as the graph with vertex set Zq and where two vertices are adjacent if their Hamming distance is at least d. The chromatic number of these graphs is presented for various sets of parameters (q, n, d). For the 4-colorings of the graphs H2(n, n − 1) a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness...

This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same ...

The visibility graph V(P ) of a point set P ⊆ R has vertex set P , such that two points v, w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P ). We characterise the 2and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main r...

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Cla...

The vertex arboricity of graph G is the minimum number of colors that can be used to color the vertices of G so that each color class induces an acyclic subgraph of G. We prove results such as this: if a connected graph G is neither a cycle nor a clique, then there is a coloring of V(G/ with at most [-A(G)/2 ~ colors, such that each color class induces a forest and one of those induced forests ...