On graphs with no induced five‐vertex path or paraglider

Abstract

Given two graphs H 1 and H 2, a graph is ( H 1 , H 2 )‐free if it contains no induced subgraph isomorphic to H 1 or H 2. For a positive integer t, P t is the chordless path on t vertices. A paraglider is the graph that consists of a chorless cycle C 4 plus a vertex adjacent to three vertices of the C 4. In this paper, we study the structure of ( P 5, paraglider)‐free graphs, and show that every such graph G satisfies χ MathClass-open( G MathClass-close) ≤ true⌈ 3 2 ω MathClass-open( G MathClass-close) true⌉, where χ ( G ) and ω ( G ) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the ( P 5, paraglider)‐free graphs G that satisfies χ ( G ) > 3 2 ω ( G ). We also construct an infinite family of ( P 5, paraglider)‐free graphs such that every graph G in the family has χ MathClass-open( G MathClass-close) = true⌈ 3 2 ω MathClass-open( G MathClass-close) true⌉ − 1. This shows that our upper bound is optimal up to an additive constant and that there is no ( 3 2 − ϵ )‐approximation algorithm for the chromatic number of ( P 5, paraglider)‐free graphs for any ϵ > 0.

Clique‐width: Harnessing the power of atoms

“…Brandstädt and Hoàng [11] showed that $$({P}_{5},\bar{{P}_{2}+{P}_{3}})$$‐free atoms with no dominating vertices and no vertex pairs $$\{x,y\}$$ with $$N(x)\subseteq N(y)$$ are either isomorphic to some specific graph $${G}^{* }$$ or all their induced $${C}_{5}$$s are dominating. Recently, Huang and Karthick [49] proved a more refined decomposition. However, it is not clear how to use these results to prove boundedness of clique‐width of $$({P}_{5},\bar{{P}_{2}+{P}_{3}})$$‐free atoms, and additional insights are needed.…”

“…Brandstädt and Hoàng [11] showed that P P P ( , + ) are either isomorphic to some specific graph G* or all their induced C 5 s are dominating. Recently, Huang and Karthick [49] proved a more refined decomposition. However, it is not clear how to use these results to prove boundedness of clique-width of P P P ( , + ) 5 2 3 -free atoms, and additional insights are needed.…”

“…This also holds, for instance, for its subclass of $$({C}_{4},2{P}_{2})$$‐free graphs and thus for $$({C}_{4},{P}_{5})$$‐free graphs and $$({C}_{4},{P}_{2}+{P}_{3})$$‐free graphs. We determine a new, incomparable case where we forbid $$2{P}_{2}$$ and $$\bar{{P}_{2}+{P}_{3}}$$ (also known as the paraglider [49]); see Figure 1 for illustrations of these forbidden induced subgraphs.…”

On the chromatic number of some $P_5$-free graphs

“…2 . Huang and Karthick [18] showed that if G is (P 5 , paraglider)-free, then χ(G) ≤ ⌈ 3ω(G) 2 ⌉. Chudnovsky and Sivaraman [7] showed that χ(G) ≤ 2 ω(G)−1 if G is (P 5 , C 5 )-free, Brause et al [1] proved that χ(G) ≤ d • ω 3 (G) for some constant d if G is (P 5 , K 2,3 )-free, and Schiermeyer [21]…”

The chromatic number of (P_5, HVN )-free graphs

“…In the same paper, they further showed that if G is (P 5 , K 1 + (K 1 ∪ K 3 ))free, then χ(G) ≤max{2ω(G), 15}, and they construct an infinite family of (P 5 , K 1 +(K 1 ∪K 3 ))-free graphs such that every graph G in the family satisfies χ(G) = 2ω(G). Huang and Karthick [17] showed that if G is (P 5 , paraglider)-free, then χ(G) ≤ ⌈ 3ω(G) 2 ⌉. Char and Karthick [2] proved that a (P 5 , 4-wheel)-free graph G is 3 2 ω(G)-colorable.…”