It has been shown that every quadrangulation on any nonspherical orientable closed surface with a suf®ciently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N k has chromatic number at least 4 if G has a cycle of odd length which cuts open N k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity.
In this paper, we shall show that an irreducible triangulation of a closed surface F2 has at most cg vertices, where g stands for a genus of F2 and c a constant. 0 1995 John Wiley & Sons, Inc.
Let p be a set of connected graphs. An p -factor of a graph is its spanning subgraph such that each component is isomorphic to one of the members in p . Let P k denote the path of order k. Akiyama and Kano have conjectured that every 3-connected cubic graph of order divisible by 3 has a fP 3 g-factor. Recently, Kaneko gave a necessary and suf®cient condition for a graph to have a fP 3 , P 4 , P 5 g-factor. As a corollary, he proved that every cubic graph has a fP 3 , P 4 , P 5 g-factor. In this paper, we prove that every 2-connected cubic graph of order at least six has a fP k j k ! 6g-factor, and hence has a fP 3 , P 4 g-factor. ß
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