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arXiv · 1403.5807

Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results

Armen S. Asratian·Carl Johan Casselgren
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Abstract

Let $G$ be a Class 1 graph with maximum degree $4$ and let $t\geq 5$ be an integer. We show that any proper $t$-edge coloring of $G$ can be transformed to any proper $4$-edge coloring of $G$ using only transformations on $2$-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper $5$-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent.

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Armen S. Asratian, Carl Johan Casselgren. 2014-03-23. Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results. https://arxiv.org/abs/1403.5807

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