Colored graphs without colorful cycles

13 years 4 months ago

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A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a diﬀerent color. We ﬁrst show that a colored graph lacks colorful cycles iﬀ it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation m◦n ≡ m+n−2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two diﬀerent ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The ﬁrst characterization...

Richard N. Ball, Ales Pultr, Petr Vojtechovsk&yacu

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