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Defective Ramsey Numbers in Graph Classes by Demirci, Yunus Emre; Ekim, Tınaz; Gimbel, John; Yıldız, Mehmet Akif is a scholarly article available to read on EtoBox.

What is Defective Ramsey Numbers in Graph Classes about?

Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$ vertices that is $k$-sparse in the complement of $G$. As a generalization of Ramsey numbers, the $k$-defective Ramsey number $R_k^{\mathcal{G}}(i,j)$ for the graph class $\mathcal{G}$ is defined as the smallest natural number $n$ such that all graphs on $n$ vertices in the class $\mathcal{G}$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine $R_k^{\mathcal{G}}(i,j)$ where $\mathcal{G}$ represents various graph classes. In forests and cographs, we give formulas for all defective Ramsey numbers. In cacti, bipartite graphs and split graphs, we provide defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.

Author

Demirci, Yunus Emre; Ekim, Tınaz; Gimbel, John; Yıldız, Mehmet Akif

Published

2019

Language

EN