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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

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.

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)

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@InProceedings{malafiejska_et_al:LIPIcs.ISAAC.2021.26, author = {Ma{\l}afiejska, Anna and Ma{\l}afiejski, Micha{\l} and Ocetkiewicz, Krzysztof M. and Pastuszak, Krzysztof}, title = {{Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {26:1--26:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.26}, URN = {urn:nbn:de:0030-drops-154596}, doi = {10.4230/LIPIcs.ISAAC.2021.26}, annote = {Keywords: interval edge coloring, biregular graphs, coloring algorithm} }

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