Coloring_of_some_crown-free_graphs
Di Wu, Baogang Xu

TL;DR
This paper establishes upper bounds on the chromatic number of crown-free graphs with additional forbidden subgraphs, advancing understanding of coloring properties in specific graph classes.
Contribution
It provides new upper bounds on the chromatic number for crown-free graphs excluding certain subgraphs, extending previous coloring theory results.
Findings
Proves $oxed{ ext{chromatic number} \\le rac{3}{2}( ext{clique number}^2 - ext{clique number})}$ for (crown, $P_5$)-free graphs.
Establishes $oxed{ ext{chromatic number} \\le rac{1}{2}( ext{clique number}^2 + ext{clique number})}$ for (crown, fork)-free graphs.
Shows $oxed{ ext{chromatic number} \\le rac{1}{2} ext{clique number}^2 + rac{3}{2} ext{clique number} + 1}$ for (crown, $P_3 old P_2$)-free graphs.
Abstract
Let and be two vertex disjoint graphs. The {\em union} is the graph with and . The {\em join} is the graph with and . We use to denote a {\em path} on vertices, use {\em fork} to denote the graph obtained from by subdividing an edge once, and use {\em crown} to denote the graph . In this paper, we show that (\romannumeral 1) if is (crown, )-free, (\romannumeral 2) if is (crown, fork)-free, and (\romannumeral 3) if is (crown, )-free.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Graph theory and applications
