5-Coloring reconfiguration of planar graphs with no short odd cycles

Pub Date : 2023-12-06 DOI:10.1002/jgt.23064

Daniel W. Cranston, Reem Mahmoud

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Abstract

The coloring reconfiguration graph $C_{k} (G)$ has as its vertex set all the proper $k$ -colorings of $G$ , and two vertices in $C_{k} (G)$ 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 $C_{k} (G)$ is $O (n^{2})$ . Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $C_{5} (G)$ 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 $C_{5} (G)$ is $O (n^{2})$ for every planar graph $G$ with no 3-cycles and no 5-cycles.

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5 无奇数短周期平面图的着色重构

着色重构图 Ck(G)${{{mathscr{C}}}_{k}(G)$ 的顶点集是 G$G$ 的所有适当 k$k$ 着色，如果 Ck(G)${{{mathscr{C}}}_{k}(G)$ 中的两个顶点在一个顶点上的 k$k$ 着色不同，则这两个顶点相邻。塞雷塞达猜想，如果一个 n$n$ 个顶点的图 G$G$ 是 d$d$ 退化的，并且 k≥d+2$k\ge d+2$，那么 Ck(G)${{{mathscr{C}}_{k}(G)$ 的直径是 O(n2)$O({n}^{2})$。布斯凯与海因里希证明，如果 G$G$ 是平面且双向的，那么 C5(G)${{{mathscr{C}}_{5}(G)$ 的直径是 O(n2)$O({n}^{2})$（这证明了对每一个退化度为 3 的此类图的塞雷塞达猜想。作为这个问题的部分解决方案，我们证明了对于每一个没有 3 循环和 5 循环的平面图 G$G$ ，C5(G)${{{mathscr{C}}}_{5}(G)$ 的直径都是 O(n2)$O({n}^{2})$ 。

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