Abstract
It is well known that if the edges of a finite simple connected graph on n vertices are colored so that no cycle is rainbow, then no more than n-1 colors can appear on the edges. In previous work it has been shown that the essentially different rainbow-cycle-forbidding edge colorings of Kn with n-1 colors appearing are in 1-1 correspondence with (can be encoded by) the (isomorphism classes of) full binary trees with n leafs. In the encoding, the natural Huffman labeling of each tree arising from the assignment of 1 to each leaf plays a role. Very recently it has been shown that a similar encoding holds for rainbow-cycle-forbidding edge colorings of Ka,b with a+b-1 colors appearing. In this case the binary trees are given Huffman labelings arising from certain assignments of (0,1) or (1,0) to the leafs. (Sibling leafs are not allowed to be assigned the same label.) In this paper we prove the analogous result for complete r-partite graphs, for r > 2.
Highlights
Edge colorings of Kn which forbid rainbow cycles, Utilitas Mathematica, 83 (2010), 219-232.
Edge colorings of Km,n with m + n − 1 colors which forbid rainbow cycles, Theory and Applications of Graphs, 4 (2017), issue 1, Article 1, 17 pp.
G.A. Harris, and P.D. Johnson, Introduction to Information Theory, 2nd edition, Chapman & Hall/CRC, Boca Raton, 2003
Summary
Edge colorings of Kn which forbid rainbow cycles, Utilitas Mathematica, 83 (2010), 219-232.
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