Kempe changes in $H$-free graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908681945219072 |
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| author | Belavadi, Manoj Cameron, Kathie |
| author_facet | Belavadi, Manoj Cameron, Kathie |
| contents | Given a $k$-colouring of a graph $G$ and two of the colours, a $Kempe$ $chain$ is a connected component of the subgraph of $G$ induced by the vertices coloured with one of these two colours. A $Kempe$ $swap$ changes one colouring into another by interchanging the colours of the vertices in a Kempe chain. Two colourings are $Kempe$ $equivalent$ if each can be obtained from the other by a series of Kempe swaps; the set of Kempe equivalent colourings is called a $Kempe$ $class$. For a graph $G$, let $χ(G)$ denote its chromatic number and let $\mathcal{C}_{k}(G)$ denote the set of all $k$-colourings of $G$. We say $G$ is $Kempe$ $connected$ if for all $k\ge χ(G)$, $\mathcal{C}_{k}(G)$ forms a Kempe class. For a graph $H$, graph $G$ is called $H$-$free$ if no induced subgraph of $G$ is isomorphic to $H$.
We prove that every $H$-free graph is Kempe connected if and only if $H$ is an induced subgraph of the path on four vertices, $P_4$.
The graph 2$K_2$ consists of four vertices and two edges which are not adjacent. We prove that for all $p\ge 0$, there is a $k$-colourable 2$K_2$-free graph $G$ such that $\mathcal{C}_{k+p}(G)$ does not form a Kempe class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00695 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kempe changes in $H$-free graphs Belavadi, Manoj Cameron, Kathie Combinatorics 05C15 Given a $k$-colouring of a graph $G$ and two of the colours, a $Kempe$ $chain$ is a connected component of the subgraph of $G$ induced by the vertices coloured with one of these two colours. A $Kempe$ $swap$ changes one colouring into another by interchanging the colours of the vertices in a Kempe chain. Two colourings are $Kempe$ $equivalent$ if each can be obtained from the other by a series of Kempe swaps; the set of Kempe equivalent colourings is called a $Kempe$ $class$. For a graph $G$, let $χ(G)$ denote its chromatic number and let $\mathcal{C}_{k}(G)$ denote the set of all $k$-colourings of $G$. We say $G$ is $Kempe$ $connected$ if for all $k\ge χ(G)$, $\mathcal{C}_{k}(G)$ forms a Kempe class. For a graph $H$, graph $G$ is called $H$-$free$ if no induced subgraph of $G$ is isomorphic to $H$. We prove that every $H$-free graph is Kempe connected if and only if $H$ is an induced subgraph of the path on four vertices, $P_4$. The graph 2$K_2$ consists of four vertices and two edges which are not adjacent. We prove that for all $p\ge 0$, there is a $k$-colourable 2$K_2$-free graph $G$ such that $\mathcal{C}_{k+p}(G)$ does not form a Kempe class. |
| title | Kempe changes in $H$-free graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2512.00695 |