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Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees

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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.

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Kategoria:
Aktywność konferencyjna
Typ:
publikacja w wydawnictwie zbiorowym recenzowanym (także w materiałach konferencyjnych)
Język:
angielski
Rok wydania:
2021
Opis bibliograficzny:
Małafiejska A., Małafiejski M., Ocetkiewicz K., Pastuszak K.: Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees// / : , 2021,
DOI:
Cyfrowy identyfikator dokumentu elektronicznego (otwiera się w nowej karcie) 10.4230/lipics.isaac.2021.26
Weryfikacja:
Politechnika Gdańska

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