Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees

Authors Anna Małafiejska, Michał Małafiejski, Krzysztof M. Ocetkiewicz, Krzysztof Pastuszak



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Author Details

Anna Małafiejska
  • Independent Researcher, Gdańsk, Poland
Michał Małafiejski
  • Faculty of Electronics, Telecommunications and Informatics, Gdańsk Tech, Poland
  • Faculty of Business and Technologies, International Black Sea University, Tbilisi, Georgia
Krzysztof M. Ocetkiewicz
  • CI TASK, Gdańsk Tech, Poland
Krzysztof Pastuszak
  • Faculty of Electronics, Telecommunications and Informatics, Gdańsk Tech, Poland

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Anna Małafiejska, Michał Małafiejski, Krzysztof M. Ocetkiewicz, and Krzysztof Pastuszak. Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.26

Abstract

An edge coloring of a graph G is called interval edge coloring if for each v ∈ V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by χ'_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (α,β)-biregular if each vertex in one part has degree α and each vertex in the other part has degree β. A graph G is called (α*,β*)-bipartite if G is a subgraph of an (α,β)-biregular graph and the maximum degree in one part is α and the maximum degree in the other part is β. In the paper we study the problem of interval edge colorings of (k*,2*)-bipartite graphs, for k ∈ {3,4,5}, and of (5*,3*)-bipartite graphs. We prove that every (5*,2*)-bipartite graph admits an interval edge coloring using at most 6 colors, which can be found in O(n^{3/2}) time, and we prove that an interval edge 5-coloring of a (5*,2*)-bipartite graph can be found in O(n^{3/2}) time, if it exists. We show that every (4^*,2^*)-bipartite graph admits an interval edge 4-coloring, which can be found in O(n) time. The two following problems of interval edge coloring are known to be NP-complete: 6-coloring of (6,3)-biregular graphs (Asratian and Casselgren (2006)) and 5-coloring of (5*,5*)-bipartite graphs (Giaro (1997)). In the paper we prove NP-completeness of 5-coloring of (5*,3*)-bipartite graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
Keywords
  • interval edge coloring
  • biregular graphs
  • coloring algorithm

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References

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