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Some results on cyclic interval edge colorings of graphs

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#### Co-author(s) :

Casselgren Carl Johan ; Petrosyan Petros

#### Uncontrolled Keywords:

bipartite graph ; biregular graph ; complete multipartite graph ; cyclic interval coloring ; edge coloring ; interval coloring

#### 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 of even maximum degree ^
(G) ≥ 4 admits a cyclic interval ^ (G)-coloring if for every vertex v the degree dG (ν) satisfies either dG(ν) ≥ ^(G) -2 or dG(ν) < 2. We also prove that every Eulerian bipartite graph G with maximum degree at most eight 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 proves a conjecture of Petrosyan and Mkhitaryan.

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#### Affiliation:

Linköping University ; Department of Mathematics ; Yerevan State University ; Department of Informatics and Applied Mathematics