Mathematics

CLC number: O157.5

Nonisomorphic Orientable Quadrangular Embeddings and Edge-Colorings of K_12s+9

K_12s+9的不同构可定向四边形嵌入及染色

¹ College of Science, Minzu University of China, Beijing 100081, China
² Department of Basic Courses, Rocket Force University of Engineering, Xi'an 710025, Shaanxi, China

Received: 20 February 2024

Abstract

In this paper, by constructing the current graph of the complete graph $K_{\mathrm{12}s+\mathrm{9}}$ and a mapping function, we prove that $K_{\mathrm{12}s+\mathrm{9}}$ ($s$ is an odd number) has at least ${\mathrm{6}}^{\mathrm{2}s}\times {\mathrm{3}}^{\frac{s+\mathrm{3}}{\mathrm{2}}}$ nonisomorphic orientable quadrangular embeddings, and the orientable genus is $\frac{(\mathrm{12}s+\mathrm{9})(\mathrm{12}s+\mathrm{4})}{\mathrm{8}}+\mathrm{1}$. Every one of the nonisomorphic orientable quadrangular embeddings has at least twenty-four 4-edge-colors, and each color appears around each face of orientable quadrangular embeddings.

摘要

本文通过构造完全图$K_{\mathrm{12}s+\mathrm{9}}$的电流图和一个同构映射，证明了$K_{\mathrm{12}s+\mathrm{9}}$($s$是一个奇数)至少有${\mathrm{6}}^{\mathrm{2}s}\times {\mathrm{3}}^{\frac{s+\mathrm{3}}{\mathrm{2}}}$个不同构可定向四边形嵌入，且可定向亏格为$\frac{(\mathrm{12}s+\mathrm{9})(\mathrm{12}s+\mathrm{4})}{\mathrm{8}}+\mathrm{1}$。每一个不同构可定向四边形嵌入至少有24个4边染色，每种颜色出现在四边形嵌入的每一个面上。