On the chromatic number of $$P_5$$ P 5 -free graphs with no large intersecting cliques

Weilun Xu and Xia Zhang ()
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Weilun Xu: Shandong Normal University
Xia Zhang: Shandong Normal University

Journal of Combinatorial Optimization, 2023, vol. 46, issue 3, No 5, 12 pages

Abstract: Abstract A graph G is called $$(H_1, H_2)$$ ( H 1 , H 2 ) -free if G contains no induced subgraph isomorphic to $$H_1$$ H 1 or $$H_2$$ H 2 . Let $$P_k$$ P k be a path with k vertices and $$C_{s,t,k}$$ C s , t , k ( $$s\le t$$ s ≤ t ) be a graph consisting of two intersecting complete graphs $$K_{s+k}$$ K s + k and $$K_{t+k}$$ K t + k with exactly k common vertices. In this paper, using an iterative method, we prove that the class of $$(P_5,C_{s,t,k})$$ ( P 5 , C s , t , k ) -free graphs with clique number $$\omega $$ ω has a polynomial $$\chi $$ χ -binding function $$f(\omega )=c(s,t,k)\omega ^{\max \{s,k\}}$$ f ( ω ) = c ( s , t , k ) ω max { s , k } . In particular, we give two improved chromatic bounds: every $$(P_5, butterfly)$$ ( P 5 , b u t t e r f l y ) -free graph G has $$\chi (G)\le \frac{3}{2}\omega (G)(\omega (G)-1)$$ χ ( G ) ≤ 3 2 ω ( G ) ( ω ( G ) - 1 ) ; every $$(P_5, C_{1,3})$$ ( P 5 , C 1 , 3 ) -free graph G has $$\chi (G)\le 9\omega (G)$$ χ ( G ) ≤ 9 ω ( G ) .

Keywords: $$P_5$$ P 5 -free graph; Chromatic number; Polynomial $$\chi $$ χ -boundedness; Intersecting cliques; 05C15 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10878-023-01088-5

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