Abstract
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of G. Scott [Graphs and Combinatorics, 2001] proved that a graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G is k-odd colourable if it can be partitioned into at most k odd induced subgraphs. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. In particular, we consider for a number of classes whether they have bounded odd chromatic number. Our main results are that interval graphs, graphs of bounded modular-width and graphs of bounded maximum degree all have bounded odd chromatic number.
Ararat Harutyunyan is supported by the grant from French National Research Agency under JCJC program (DAGDigDec: ANR-21-CE48-0012).
Noleen Köhler is supported by the grant from French National Research Agency under JCJC program (ASSK: ANR-18-CE40-0025-01).
Nikolaos Melissinos is partially supported by the CTU Global postdoc fellowship program.
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Notes
- 1.
While Fomin et al. proved the lower bound for clique-width, it also holds for rank-width, since rank-width is always at most clique-width.
- 2.
Subdividing an edge uv consists in removing uv, adding a new vertex w, and making it adjacent to exactly u and v.
- 3.
This definition of odd colouring is not to be confused with the one introduced by Petrusevski and Skrekovski [10], which is a specific type of proper colouring.
- 4.
For every result which is marked by \((*)\) the proof can be found in the full version of the paper.
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Belmonte, R., Harutyunyan, A., Köhler, N., Melissinos, N. (2023). Odd Chromatic Number of Graph Classes. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_4
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