文摘
A Grünbaum coloring of a triangulation g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si1.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=2ef60b2b59b6e5cbf989d6dac5195bed" title="Click to view the MathML source">G on a surface is a 3-edge coloring of g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si1.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=2ef60b2b59b6e5cbf989d6dac5195bed" title="Click to view the MathML source">G such that each face of g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si1.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=2ef60b2b59b6e5cbf989d6dac5195bed" title="Click to view the MathML source">G receives three distinct colors on its boundary edges. In this paper, we prove that every Fisk triangulation on the projective plane g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si18.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=e8286c3bbbcb8b768cf8891ea7ac6b95" title="Click to view the MathML source">P has a Grünbaum coloring, where a “Fisk triangulation” is one with exactly two odd degree vertices such that the two odd vertices are adjacent. To prove the theorem, we establish a generating theorem for Fisk triangulations on g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si18.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=e8286c3bbbcb8b768cf8891ea7ac6b95" title="Click to view the MathML source">P. Moreover, we show that a triangulation g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si1.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=2ef60b2b59b6e5cbf989d6dac5195bed" title="Click to view the MathML source">G on g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si18.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=e8286c3bbbcb8b768cf8891ea7ac6b95" title="Click to view the MathML source">P has a Grünbaum coloring with each color-induced subgraph connected if and only if every vertex of g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303328&_mathId=si1.gif&_user=111111111&_pii=S0166218X16303328&_rdoc=1&_issn=0166218X&md5=2ef60b2b59b6e5cbf989d6dac5195bed" title="Click to view the MathML source">G has even degree.