Recoloring Planar Graphs of Girth at Least Five

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

“…to every incident vertex. 1 (R2): Every 3-vertex not in T takes 1 3 from every neighbor that has level at least 3.…”

“…. Subsequently, many results [1,3,8,10], as shown in give an exponential upper bound on its diameter. Bartier et al [1] proved that…”

“…Cereceda, van den Heuvel, and Johnson [5, 6] were the first to study the connectedness of $${{\mathscr{C}}}_{k}(G)$$. Subsequently, many results [1, 3, 8, 10], as shown in Table 1, have focused on the diameter of $${{\mathscr{C}}}_{k}(G)$$; in particular, these papers attempt to tackle Cereceda's Conjecture [4] that diam$$({{\mathscr{C}}}_{k}(G))=O({n}^{2})$$ whenever $$G$$ is $$d$$‐degenerate and $$k\ge d+2$$. (The bound $$O({n}^{2})$$ is best possible, as shown by Bonamy et al [2]) Most results on the connectedness of $${{\mathscr{C}}}_{k}(G)$$ give an exponential upper bound on its diameter.…”

“…(The bound $$O({n}^{2})$$ is best possible, as shown by Bonamy et al [2]) Most results on the connectedness of $${{\mathscr{C}}}_{k}(G)$$ give an exponential upper bound on its diameter. Bartier et al [1] proved that $$\,\text{diam}\,({{\mathscr{C}}}_{5}(G))=O(n)$$ for every planar graph $$G$$ of girth at least 6, while Dvořák and Feghali [8] proved $$\,\text{diam}\,({{\mathscr{C}}}_{7}(G))=O(n)$$ for every triangle‐free planar graph $$G$$. Feghali [10] showed that if $$d\ge 1$$ and $$k\ge d+1$$, then for every $$\epsilon \gt 0$$ and every graph $$G$$ with maximum average degree at most $$d-\epsilon $$, we have $$\,\text{diam}\,({{\mathscr{C}}}_{k}(G))=O(n{(\mathrm{log}n)}^{d-1})$$…”