外1-平面图的均匀边染色

图[G]的[s]-均匀边[k]-染色是指用[k]种颜色对图的边进行染色，使得图[G]的每个顶点所关联的任何两种颜色的边的条数至多相差[s]。使得对于每个不小于[k]的整数[t]，图[G]都具有[s]-均匀边[t]-染色的最小整数[k]称为图[G]的[s]-均匀边色数阈值。文中证明了外1-平面图的1-均匀边色数阈值最多为5，不含有相邻的3圈的外1-平面图的均匀边色数阈值最多为4，外1-平面图的2-均匀边色数阈值恰好为1。

Equitable Edge Coloring of Outer-1-Planar Graphs

An [s]-equitable edge-[k]-coloringof a graph [G] is an edge coloring of [G] using [k] colors so that the sizes of any two color classes incident with any fixed vertex of [G] differ by at most [s]. The [s]-equitable edge chromatic threshold of [G] is the smallest [k] such that [G] has an [s]-equitable edge-[t]-colorings for integer [t] that is no less than [k]. It is proved that the 1-equitable edge chromatic threshold of any outer-1-planar graph is at most 5, the 1-equitable edge chromatic threshold of any outer-1-planar graph without adjacent triangles is at most 4, and the 2-equitable edge chromatic threshold of any outer-1-planar graph is exactly 1.