Odd chromatic number of graph classes

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F25%3A00380477" target="_blank" >RIV/68407700:21240/25:00380477 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1002/jgt.23200" target="_blank" >https://doi.org/10.1002/jgt.23200</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/jgt.23200" target="_blank" >10.1002/jgt.23200</a>

Result language

angličtina

Original language name

Odd chromatic number of graph classes

Original language description

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 $G$. Scott proved that a connected graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G $G$ is k $k$-odd colourable if it can be partitioned into at most k $k$ odd induced subgraphs. The odd chromatic number of G $G$, denoted by chi odd( G ) ${chi }_{text{odd}}(G)$, is the minimum integer k $k$ for which G $G$ is k $k$-odd colourable. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. We first consider a question due to Scott, which states that every graph G $G$ of even order n $n$ has chi odd( G ) <= c n ${chi }_{text{odd}}(G)le csqrt{n}$, for some positive constant c $c$, by proving that this is indeed the case if G $G$ is restricted to having girth at least seven. We also show that any graph G $G$ whose all components have even order satisfies chi odd( G ) <= 2 Delta - 1 ${chi }_{text{odd}}(G)le 2{rm{Delta }}-1$, where Delta ${rm{Delta }}$ is the maximum degree of G $G$. Next, we show that certain interesting classes have bounded odd chromatic number. Our main results in this direction are that interval graphs, graphs of bounded modular-width all have bounded odd chromatic number. In particular, every even interval graph is 6-odd colourable, and every even graph is 3 m w $3mw$-odd colourable, where m w $mw$ is the modular width of a graph.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Journal of Graph Theory

ISSN

0364-9024

e-ISSN

1097-0118

Volume of the periodical

108

Issue of the periodical within the volume

4

Country of publishing house

GB - UNITED KINGDOM

Number of pages

23

Pages from-to

722-744

UT code for WoS article

001357390600001

EID of the result in the Scopus database

2-s2.0-85208258355