Grünbaum colorings of triangulations on the projective plane

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• 作者：Michiko Kasai^a ; ^{mikko2046@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Naoki Matsumoto^b ; ^{naoki-matsumoto@st.seikei.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Atsuhiro Nakamoto^a ; ^{nakamoto@ynu.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Grü ; nbaum coloring ; Triangulation ; Projective plane
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 December 2016
• 年：2016
• 卷：215
• 期：Complete
• 页码：155-163
• 全文大小：466 K

文摘

A Grünbaum coloring   of a triangulation G$G$ on a surface is a 3-edge coloring of G$G$ such that each face of G$G$ receives three distinct colors on its boundary edges. In this paper, we prove that every Fisk triangulation   on the projective plane bbbcb8b768cf8891ea7ac6b95" title="Click to view the MathML source">P$\mathbb{P}$ 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 bbbcb8b768cf8891ea7ac6b95" title="Click to view the MathML source">P$\mathbb{P}$. Moreover, we show that a triangulation G$G$ on bbbcb8b768cf8891ea7ac6b95" title="Click to view the MathML source">P$\mathbb{P}$ has a Grünbaum coloring with each color-induced subgraph connected if and only if every vertex of G$G$ has even degree.