5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles
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arXiv
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| Natura: | Preprint |
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2022
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| author | Cranston, Daniel W. Mahmoud, Reem |
| author_facet | Cranston, Daniel W. Mahmoud, Reem |
| contents | The coloring reconfiguration graph $\mathcal{C}_k(G)$ has as its vertex set all the proper $k$-colorings of $G$, and two vertices in $\mathcal{C}_k(G)$ are adjacent if their corresponding $k$-colorings differ on a single vertex. Cereceda conjectured that if an $n$-vertex graph $G$ is $d$-degenerate and $k\geq d+2$, then the diameter of $\mathcal{C}_k(G)$ is $O(n^2)$. Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$ for every planar graph $G$ with no 3-cycles and no 5-cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02228 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | 5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles Cranston, Daniel W. Mahmoud, Reem Combinatorics 05C15, 05C85, 05C12 The coloring reconfiguration graph $\mathcal{C}_k(G)$ has as its vertex set all the proper $k$-colorings of $G$, and two vertices in $\mathcal{C}_k(G)$ are adjacent if their corresponding $k$-colorings differ on a single vertex. Cereceda conjectured that if an $n$-vertex graph $G$ is $d$-degenerate and $k\geq d+2$, then the diameter of $\mathcal{C}_k(G)$ is $O(n^2)$. Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$ for every planar graph $G$ with no 3-cycles and no 5-cycles. |
| title | 5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles |
| topic | Combinatorics 05C15, 05C85, 05C12 |
| url | https://arxiv.org/abs/2208.02228 |