Choosability of Toroidal Graphs with Forbidden Structures

Ilkyoo Choi

KAIST

KAIST

2014/07/07 Monday 4PM-5PM

Room 1409

Room 1409

The choosability \(\chi_\ell(G)\) of a graph G is the minimum k such that having k colors available at each vertex guarantees a proper coloring. Given a toroidal graph G, it is known that \(\chi_\ell(G)\leq 7\), and \(\chi_\ell(G)=7\) if and only if G contains \(K_7\). Cai, Wang, and Zhu proved that a toroidal graph G without 7-cycles is 6-choosable, and \(\chi_\ell(G)=6\) if and only if G contains \(K_6\). They also prove that a toroidal graph G without 6-cycles is 5-choosable, and conjecture that \(\chi_\ell(G)=5\) if and only if G contains \(K_5\). We disprove this conjecture by constructing an infinite family of non-4-colorable toroidal graphs with neither \(K_5\) nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither \(K^-_5\) (a \(K_5\) missing one edge) nor 6-cycles are 4-choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4-choosable.

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