Abstract
Two vertices and in a nontrivial connected graph are twins if and have the same neighbors in . If and are adjacent, they are referred to as true twins, while if and are nonadjacent, they are false twins. For a positive integer , let be a vertex coloring where adjacent vertices may be assigned the same color. The coloring induces another vertex coloring defined by for each , where is the closed neighborhood of . Then is called a closed modular -coloring if in for all pairs , of adjacent vertices that are not true twins. The minimum for which has a closed modular -coloring is the closed modular chromatic number of . A rooted tree of order at least 3 is even if every vertex of has an even number of children, while is odd if every vertex of has an odd number of children. It is shown that for each even rooted tree and if is an odd rooted tree having no vertex with exactly one child. Exact values are determined for several classes of odd rooted trees .
Citation
Bryan Phinezy. Ping Zhang. "On closed modular colorings of rooted trees." Involve 6 (1) 83 - 97, 2013. https://doi.org/10.2140/involve.2013.6.83
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