Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency

Abstract

We call a graph (m, k)-colorable if its vertices can be colored with m colors in such a way that each vertex is adjacent to at most k vertices of the same color as itself. For the class of planar graphs, and the class of outerplanar graphs, we determine all pairs (m, k) such that every graph in the class is ( m , k)-colorable. We include an elementary proof (not assuming the truth of the four-color theorem) that every planar graph is (4, 1)-colorable. Finally, we prove that, for each compact surface S, there is an integer k = k(S) such that every graph in S can be (4, &colored; we conjecture that 4 can be replaced by 3 in this statement.

Vertex decompositions of sparse graphs into an independent vertex set and a subgraph of maximum degree at most 1

“…(0, 0, 0, 0)-colorable. Cowen, Cowen, and Woodall [3] proved that every planar graph is 2-improperly 3-colorable, i.e. (2,2,2)-colorable.…”

“…(3). Therefore, we have h(v) = 2 by (4), and let v be adjacent to the 3 + -vertices x and y (see Fig.…”

Dynamic F-free Coloring of Graphs

“…Since this subject is too wide to be surveyed in a short paper, we mention just a few examples like subcoloring known also as P 3 -free coloring (where P p denotes the chordless path on p vertices), P 4 -free coloring and improper coloring, and we refer to appropriate literature on other variants, e.g., many results on subcoloring can be found in Albertson et al [2], Broere and Mynhardt [8], Fiala et al [16] as well as in work of Gimbel and Hartman [17]. For results on P 4 -free coloring see, e.g., Gimbel and Nešetřil [18] and a paper of Hoàng and Le [23], while for improper coloring we refer the reader to papers of Bermond et al [4], Cowen et al [15] and Havet et al [21]. Concerning the computational complexity of F-free k-coloring problem we mention the result of Achlioptas [1] who proved that for any fixed graph F, except K 2 , the problem of deciding if a given graph admits an F-free coloring with at most k-colors is NP-complete (for a detailed study of the computational complexity of many variants of offline generalized colorings see, e.g., Broersma et al [9]).…”

The Relaxed Edge-Coloring Game and k-Degenerate Graphs

“…We first consider a variation of this game in which the players are creating a relaxed coloring, also called defect coloring, of the the graph G. This variation was introduced by Chou et al [4] and builds on the known work [5,6,7,14,24] concerning defect colorings of graphs. The only difference between this game and the original version is which colors the players are allowed to use.…”