Year: 2025
Type: article
Abstract: Let G be a simple connected graph, and k be a positive integer. The k th graph power $$G^k$$ G k of G is the graph whose vertex set is the vertex set of G and two distinct vertices are adjacent if and only if their distance in G is at most k . The independence number of $$G^k$$ G k is the maximum size of an independent set in $$G^k$$ G k . The chromatic number of $$G^k$$ G k , $$\chi (G^k)$$ χ ( G k ) , is the smallest number of colors needed to color $$G^k$$ G k so that the vertices sharing the same color form an independent set of $$G^k$$ G k . In this paper, we study the independence number of $$G^k$$ G k , when G is a tree, by giving a known lower bound for the number of vertices that a tree must have, assuming a certain independence number of $$G^k$$ G k . As a consequence of our proof, we describe all trees that attain this lower bound. For some of the trees given G , we also describe a ( $$\chi (G^k)$$ χ ( G k ) )-coloring of $$G^k$$ G k .
Authors: Rosário Fernandes, Rúben Palma
Institution: Universidade Nova de Lisboa
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,390
OpenAlex ID: W4414154044