Decomposing complete edge-chromatic graphs and hypergraphs. Revisited

Abstract

A d -graph G = ( V ; E 1 , … , E d ) is a complete graph whose edges are colored by d colors, that is, partitioned into d subsets some of which might be empty. We say that a d -graph G is complementary connected (CC) if the complement to each chromatic component of G is connected on V . We prove that every such d -graph contains a sub- d -graph Π or Δ , where Π has four vertices and two non-empty chromatic components each of which is a P 4 , while Δ is a three-colored triangle. This statement implies that each Π - and Δ -free d -graph is uniquely decomposable in accordance with a tree T = T ( G ) whose leaves are the vertices of V and the interior vertices of T are labeled by the colors 1 , … d . Such a tree is naturally interpreted as a positional game form (with perfect information and without moves of chance) of d players I = { 1 , … , d } and n outcomes V = { v 1 , … , v n } . Thus, we get a one-to-one correspondence between these game forms and Π - and Δ -free d -graphs. As a corollary, we obtain a characterization of the normal forms of positional games with perfect information and, in case d = 2 , several characterizations of the read-once Boolean functions. These results are not new; in fact, they are 30 and, in case d = 2 , even 40 years old. Yet, some important proofs did not appear in English. Gyárfás and Simonyi recently proved a similar decomposition theorem for the Δ -free d -graphs. They showed that each Δ -free d -graph can be obtained from the d -graphs with only two non-empty chromatic components by successive substitutions. This theorem is based on results by Gallai, Lovász, Cameron and Edmonds. We obtain some new applications of these results.