Some Results on Cyclic Interval Edge Colorings of Graphs release_a5pjmipz2zft3kdh4iwvuknque

by Armen S. Asratian, Carl Johan Casselgren, Petros A. Petrosyan

Released as a article .

2017  

Abstract

A proper edge coloring of a graph G with colors 1,2,...,t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G with even maximum degree Δ(G)≥ 4 admits a cyclic interval Δ(G)-coloring if for every vertex v the degree d_G(v) satisfies either d_G(v)≥Δ(G)-2 or d_G(v)≤ 2. We also prove that every Eulerian bipartite graph G with maximum degree at most 8 has a cyclic interval coloring. Some results are obtained for (a,b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4,7)-biregular graphs as well as all (2r-2,2r)-biregular (r≥ 2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this settles in the affirmative, a conjecture of Petrosyan and Mkhitaryan.
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Date   2017-03-29
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arXiv  1606.09389v2
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