Five-coloring graphs on the Klein bottle

Abstract

We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list.

Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle

“…This implies that |V (C 3 )| = 5 and |V (C 1 )| = |V (C 2 )| = 8. However, (6) implies that C 1 has a chord, contradicting (8), (9) or (13).…”

“…By (1) they belong to R, and so we may assume that say r 2 and r 3 have degree two. The edge r 2 r 3 is not contained in any 5-cycle, as otherwise R would have a chord or an internal vertex would have two neighbors in R, contrary to (8) and (9). Let G ′ be the graph obtained from G by contracting the edge r 2 r 3 , and let R ′ be the corresponding outer cycle of G ′ .…”

“…, v 5 }. If v 2 ∈ V (R), then x 3 is internal since v 3 has only one neighbor in R by (9), and x 4 and x 5 are internal by (15) and (1). Also, not all of x 1 , x 3 , x 4 and x 5 belong to R, as no edge of T is incident with f .…”

Coloring Eulerian Triangulations of the Klein Bottle

“…Hence, we can further assume that the vertices w 1 and v 3 are not adjacent in G. Color the vertices w 1 and v 3 with the same color, the vertices w 5 and v 1 with the same color and the vertices w 2 , . .…”

Minimal Nontoroidal Graphs on Eight Vertices

“…H 的联图，其中 7 H 是由在 4 K 的两个拷贝上应用 Hajós 构造得到的，或者环面上有 11 个点的三角剖分图 ， 它是由长度为 11 的圈添加圈上的距离为 2 和 3 点之间的边得到的图。一个有趣的问题是：其它的曲面上的临界 图是否有类似的结果？Chenette、Postle、Streib、Thomas 和 Yerger [12] 、Kawarayabashi、Král、Kynčl 和 Lidický [13] 使用不同的方法得到了克莱因瓶上的 6-色临界图完全列表。 11 T 参考文献 (References)…”