Decomposing complete edge-chromatic graphs and hypergraphs. Revisited

A d-graph G = (V ;E1, . . . , Ed) is a complete graph whose edges are colored by d colors, or in other words, are partitioned into d subsets (some of which might be empty). We say that G is complementary connected if the complement to each chromatic component of G is connected on V , or in other words, if for each two vertices u,w ∈ V and color i ∈ I = {1, . . . , d} there is a path between u and w without edges of Ei. We show that every such d-graph contains a subgraph Π or ∆ , where Π has 4 vertices and 2 non-empty chromatic components each of which is a P4, while ∆ is the three-colored triangle. This simple statement implies that each Πand ∆-free d-graph is uniquely decomposable in accordance with a tree T = T (G) whose leaves are the vertices of V and other vertices of T are labeled by the colors of I. We can naturally interpret such a tree as a positional game (with perfect information and without moves of chance) of d players I = {1, . . . , d} and n outcomes V = {v1, . . . , vn}. Thus, we get a one-to-one correspondence between these games and Πand ∆-free d-graphs and, as a corollary, a characterization of the normal forms of positional games with perfect information. Another corollary of the above decomposition of d-graphs in case d = 2 is a characterization of the read-once Boolean functions. These results are not new; in fact, they are 25-35 years old. Yet, some important proofs did not appear in English. Gyárfás and Simonyi recently proved a similar decomposition theorem for ∆-free d-graph. They showed that each such d-graph can be obtained from 2-graphs by substitutions. This theorem is based on results by Gallai, Cameron and Edmonds. We get some new applications of these results.