On the b-coloring of tight graphs

详细信息    查看全文

• 作者：Mekkia Kouider ; Mohamed Zamime
• 关键词：Coloring ; b ; coloring ; b ; chromatic number ; Tight graphs
• 刊名：Journal of Combinatorial Optimization
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：33
• 期：1
• 页码：202-214
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operation Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2886
• 卷排序：33

文摘

A coloring c of a graph \(G=(V,E)\) is a b-coloring if for every color i there is a vertex, say w(i), of color i whose neighborhood intersects every other color class. The vertex w(i) is called a b-dominating vertex of color i. The b-chromatic number of a graph G, denoted by b(G), is the largest integer k such that G admits a b-coloring with k colors. Let m(G) be the largest integer m such that G has at least m vertices of degree at least \(m-1\). A graph G is tight if it has exactly m(G) vertices of degree \(m(G)-1\), and any other vertex has degree at most \(m(G)-2\). In this paper, we show that the b-chromatic number of tight graphs with girth at least 8 is at least \(m(G)-1\) and characterize the graphs G such that \(b(G)=m(G)\). Lin and Chang (2013) conjectured that the b-chromatic number of any graph in \(\mathcal {B}_{m}\) is m or \(m-1\) where \(\mathcal {B}_{m}\) is the class of tight bipartite graphs \((D,D{^\prime })\) of girth 6 such that D is the set of vertices of degree \(m-1\). We verify the conjecture of Lin and Chang for some subclass of \(\mathcal {B}_{m}\), and we give a lower bound for any graph in \(\mathcal {B}_{m}\).