b-continuity and Partial Grundy Coloring of graphs with large girth
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A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of G is the set S_b(G) of integers k such that G has a b-coloring with k colors and b(G)=max S_b(G) is the b-chromatic number of G. A graph is b-continous if S_b(G)=[χ(G),b(G)]∩Z. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. A partial Grundy coloring is a proper coloring f:V(G)→{1,...,k} such that each color class i contains some vertex u that is adjacent to every color class j such that j<i. The partial Grundy number of G is the maximum value ∂Γ(G) for which G has a partial Grundy coloring. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph G with girth at least 7 contains the integers between 2χ(G) and b(G). We also prove that ∂Γ(G) equals a known upper bound when G is a graph with girth at least 7. These results generalize previous ones by Linhares-Sales and Silva (2017), and by Shi et al.(2005).
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