The edge chromatic transformation index of graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911296361857024 |
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| author | Asratian, Armen S. Casselgren, Carl Johan |
| author_facet | Asratian, Armen S. Casselgren, Carl Johan |
| contents | Given a graph or multigraph $G$, let $χ'_{trans}(G)$ denote the minimum integer $n$ such that any proper $χ'(G)$--edge coloring of $G$ can be transformed into any other proper $χ'(G)$--edge coloring of $G$ by a series of transformations such that each of the intermediate colorings is a proper $χ'(G)$--edge coloring of $G$ and each of the transformations involves at most $n$ color classes of the previous coloring. We call $χ'_{trans}(G)$ the {\it edge chromatic transformation index of $G$}.
In this paper we show that if $G$ is a graph with maximum degree at least $4$, where every block is either a bipartite graph, a series-parallel graph, a chordless graph, a wheel graph or a planar graph of girth at least $7$, then $χ'_{trans}(G)\leq 4$. This bound is sharp for series-parallel and wheel graphs. We also show that $χ'_{trans}(G)\leq 8$ for all planar graphs $G$, $χ'_{trans}(G)\leq 5$ if $G$ is a Halin graph and $χ'_{trans}(G)=2$ if $G$ is a regular bipartite planar multigraph. Finally, we consider the analogous problem for vertex colorings, and show that for any $k\geq 3$ there is an infinite class $\cal G$$(k)$ of graphs with chromatic number $k$ such that for every $G\in \cal G$$(k)$ any two proper $k$-vertex colorings of $G$
can be transformed to each other only by a transformation, involving all $k$ color classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01614 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The edge chromatic transformation index of graphs Asratian, Armen S. Casselgren, Carl Johan Combinatorics 05C15 Given a graph or multigraph $G$, let $χ'_{trans}(G)$ denote the minimum integer $n$ such that any proper $χ'(G)$--edge coloring of $G$ can be transformed into any other proper $χ'(G)$--edge coloring of $G$ by a series of transformations such that each of the intermediate colorings is a proper $χ'(G)$--edge coloring of $G$ and each of the transformations involves at most $n$ color classes of the previous coloring. We call $χ'_{trans}(G)$ the {\it edge chromatic transformation index of $G$}. In this paper we show that if $G$ is a graph with maximum degree at least $4$, where every block is either a bipartite graph, a series-parallel graph, a chordless graph, a wheel graph or a planar graph of girth at least $7$, then $χ'_{trans}(G)\leq 4$. This bound is sharp for series-parallel and wheel graphs. We also show that $χ'_{trans}(G)\leq 8$ for all planar graphs $G$, $χ'_{trans}(G)\leq 5$ if $G$ is a Halin graph and $χ'_{trans}(G)=2$ if $G$ is a regular bipartite planar multigraph. Finally, we consider the analogous problem for vertex colorings, and show that for any $k\geq 3$ there is an infinite class $\cal G$$(k)$ of graphs with chromatic number $k$ such that for every $G\in \cal G$$(k)$ any two proper $k$-vertex colorings of $G$ can be transformed to each other only by a transformation, involving all $k$ color classes. |
| title | The edge chromatic transformation index of graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2512.01614 |