Conflict-Free Coloring of Star-Free Graphs on Open Neighborhoods
Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, Rogers Mathew

TL;DR
This paper investigates conflict-free coloring of star-free graphs on open neighborhoods, establishing bounds on the number of colors needed and demonstrating the existence of graphs requiring logarithmic colors.
Contribution
It provides upper bounds on the conflict-free chromatic number for S_k-free graphs and shows that some claw-free graphs need logarithmic colors.
Findings
h_{ON}(G) = O(k \, \log^{2+\epsilon} \Delta) for S_k-free graphs
Existence of claw-free graphs requiring ig(\, \log \Delta\big) colors
Bounds depend on maximum degree and forbidden star size
Abstract
Given a graph, the conflict-free coloring problem on open neighborhoods (CFON) asks to color the vertices of the graph so that all the vertices have a uniquely colored vertex in its open neighborhood. The smallest number of colors required for such a coloring is called the conflict-free chromatic number and denoted . In this note, we study this problem on -free graphs where is a star on vertices. When is -free, we show that , for any , where denotes the maximum degree of . Further, we show existence of claw-free (-free) graphs that require colors.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
