TY - JOUR
AU - Matsumoto, Naoki
AB - It is well-known that every triangulation on a closed surface F2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F^2$$\end{document} has a spanning quadrangulation Q, and in particular, Q is bipartite if F2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F^2$$\end{document} is the sphere. In other words, every triangulation on the sphere has a polychromatic 2-coloring, which is a (not necessarily proper) 2-coloring of a graph on a closed surface without a monochromatic face. In this paper, we consider the balancedness of a polychromatic 2-coloring of graphs, that is, whether the difference in size of each color class is at most one. We verify that every 3-colorable triangulation has a balanced polychromatic 2-coloring. On the other hand, we construct an infinite family of triangulations G with order n on the sphere such that for every polychromatic 2-coloring of G, the size of color classes differs at least n3-2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{n}{3}-2$$\end{document}. Then we conjecture that every triangulation on the sphere has a polychromatic 2-coloring whose size of color classes differs at most one-third of the number of vertices, and we give partial solutions for the conjecture.
TI - Balanced Polychromatic 2-Coloring of Triangulations
JF - Graphs and Combinatorics
DO - 10.1007/s00373-021-02420-8
DA - 2022-02-01
UR - https://www.deepdyve.com/lp/springer-journals/balanced-polychromatic-2-coloring-of-triangulations-MJweaKXaJs
VL - 38
IS - 1
DP - DeepDyve
ER -