Graph coloring is a widely used technique for allocation of time and frequency slots to nodes, for forming clusters, for constructing independent sets and dominating sets on wireless ad hoc and sensor networks. A good coloring approach should produce low color count as possible. Besides, since the nodes of a wireless ad hoc and sensor network operate with limited bandwidth, energy and computing...

An equitable coloring of a graph G is a proper coloring of the vertices of G such that color classes differ in size by at most one. In this note, we verify the equitable coloring conjecture [W. Meyer, Amer. Math. Monthly 80 (1973), 920–922] for some classes of graphs which are defined by forbidden induced subgraphs using known results.

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k > 0 colors. Each different ruleset specifies that game’s coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, an...

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ asked if can $\chi_d$-color so that the maximum degree induced by any color class is at most $\Delta^*$. We show this natural generalization of much harder on several basic classes. particular, it NP-hard split graphs, even when one parameters set to smallest possible fixed value does not trivialize problem ($\...

A. We investigate the parameterized complexity of generalisations and variations of the dominating set problem on classes of graphs that are nowhere dense. In particular, we show that the distance-d dominating-set problem, also known as the (k, d)-centres problem, is fixed-parameter tractable on any class that is nowhere dense and closed under induced subgraphs. This generalises known re...

Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic ...

Consider a graph and a collection of (not necessarily edge-disjoint) connected spanning subgraphs (factors) of the graph. We consider the problem of coloring the vertices of the graph so that each color class of the vertices dominates each factor. We find upper and lower bounds on α(t, k), which we define as the minimum radius of domination d such that every graph with a collection of k factors...

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. In this paper we study the independence, domination and chromatic numbers of 3-arc graphs ...

A proper coloring of a graph G is equitable if the sizes of any two color classes are differ by at most one. The related notion is `-bounded coloring where each of the color classes is of cardinality ≤ `. We consider the problems to determine for a given graph G (and a given integer `) whether G has an equitable (`-bounded) coloring. We prove that both problems can be solved in polynomial time ...