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Partitions, Functions and the Arc-Coloring of Digraphs Hiroyuki KAWAI Yukio SHIBATA Publication IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences Vol.E89-A No.9 pp.2381-2385 Publication Date: 2006/09/01 Online ISSN: 1745-1337 DOI: 10.1093/ietfec/e89-a.9.2381 Print ISSN: 0916-8508 Type of Manuscript: PAPER Category: Graphs and Networks Keyword: arc-coloring, set partition, line digraph, Full Text: PDF(182.6KB)>>
Summary: Let f and g be two maps from a set E into a set F such that f(x)  g(x) for every x in E. Sahili [8] has shown that, if min {|f-1(z)|,|g-1(z)|}≤ n for each z∈ F, then E can be partitioned into at most 2n+1 sets E1,..., E2n+1 such that f(Ei)∩ g(Ei)= for each i=1,..., 2n+1. He also asked if 2n+1 is the best possible bound. By using Sahili's formulation of the problem in terms of the chromatic number of line digraphs, we improve the upper bound from 2n+1 to O(log n). We also investigate extended version for more than two maps. | ||||||||||
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