Limits of near-coloring of sparse graphs

Paul Dorbec (1) , Tomas Kaiser (2) , Mickael Montassier (3) , André Raspaud (3)

1 Combinatoire et Algorithmique
2 department of mathematics and institute of computer science
3 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Let a, b, d be non-negative integers. A graph G is (d, a, b)∗ -colorable if its vertex set can be partitioned into a + b sets I1 , . . . , Ia , O1 , . . . , Ob such that the graph G[Ii ] induced by Ii has maximum degree at most d for 1 ≤ i ≤ a, while the graph G[Oj ] induced by Oj is an edgeless graph for 1 ≤ j ≤ b. In this paper, we give two real-valued functions f and g such that any graph with maximum average degree at most f(d, a, b) is (d, a, b)∗ -colorable, and there exist non-(d, a, b)∗ -colorable graphs with average degree at most g(d, a, b). Both these functions converge (from below) to 2a + b when d tends to infinity. Counterintuitively, this implies that allowing a color to be d-improper (i.e. of type Ii ) even for a large degree d increases the maximum average degree that guarantees the existence of a valid coloring only by 1. Using a color of type Ii (even with a very large degree d) is somehow less powerful than using two colors of type Oj (two stable sets).

Mots clés: colorings; improper colorings; maximum average degree

Domaines: Mathématique discrète [cs.DM]

Paul Dorbec, Tomas Kaiser, Mickael Montassier, André Raspaud. Limits of near-coloring of sparse graphs. Journal of Graph Theory, 2014, 75 (2), pp.191-202. ⟨10.1002/jgt.21731⟩. ⟨hal-00911263⟩