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Some results on cyclic interval edge colorings of graphs
Author(s) -
Asratian Armen S.,
Casselgren Carl Johan,
Petrosyan Petros A.
Publication year - 2018
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22154
Subject(s) - combinatorics , mathematics , edge coloring , bipartite graph , complete coloring , list coloring , 1 planar graph , discrete mathematics , brooks' theorem , degree (music) , indifference graph , interval graph , vertex (graph theory) , fractional coloring , interval (graph theory) , conjecture , graph coloring , graph , chordal graph , graph power , line graph , physics , acoustics
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 Δ ( G ) ≥ 4 admits a cyclic interval Δ ( G ) ‐coloring if for every vertex v the degreed G ( v )satisfies eitherd G ( v ) ≥ Δ ( G ) − 2 ord G ( v ) ≤ 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 ( 2 r − 2 , 2 r ) ‐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.