Partitioning into graphs with only small components

Partitioning H -minor free graphs into three subgraphs with no large components

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AbstractWe prove that for every graph H, if a graph G has no (odd) H minor, then its vertex set V (G) can be partitioned into three sets X 1 , X 2 , X 3 such that for each i, the subgraph induced on X i has no component of size larger than a function of H and the maximum degree of G. This improves a previous result of Alon, Ding, Oporowski and Vertigan (2003) stating that V (G) can be partitioned into four such sets if G has no H minor. Our theorem generalizes a result of Esperet and Joret (2014), who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no H minor.

As a corollary, we prove that for every positive integer t, if a graph G has no K t+1 minor, then its vertex set V (G) can be partitioned into 3t sets X 1 , .

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Islands in Graphs on Surfaces

“…We can modify the gadget J y,z,t in the proof of Theorem 11 (1), so that the same result holds for graphs with arbitrarily large (but fixed) girth. Note that in this case we lose the 2-degeneracy.…”

“…The case t = 5 was shown to be equivalent to the famous 4 Color Theorem, which states that every planar graph has a proper 4-coloring. On the other hand, it was proved by Kleinberg, Motwani, Raghavan, and Venkatasubramanian [13], and independently by Alon, Ding, Oporowski and Vertigan [1] that there is no constant c such that every planar graph has a 3-coloring in which every monochromatic component has size at most c. More generally, for every t, there are graphs with no K tminor that cannot be colored with t − 2 colors such that all monochromatic components have size bounded by a function of t (see [14]). It follows that the bound predicted by Hadwiger's conjecture (and proved for t = 5, 6) on the chromatic number of a graph with no K t -minor is best possible, even in our relaxed setting.…”

Coloring subgraphs with restricted amounts of hues

“…Most obviously, proper colorings are the case that F D K 2 and A D f2g. Thereafter, probably most studied is the case of coloring the vertices without creating some monochromatic subgraph, such as a star; these are often called defective colorings (see for example [1][2][3][4]). Defective colorings correspond to RASH colorings where A D f2; 3; : : : ; jF jg (where we use jF j to denote the order of F ).…”