On the \(e\)-positivity of \((claw, 2K_2)\)-free graphs
Summary: Motivated by Stanley and Stembridge's conjecture [\textit{R. P. Stanley} and \textit{J. R. Stembridge}, J. Comb. Theory, Ser. A 62, No. 2, 261--279 (1993; Zbl 0772.05097)] about the \(e\)-positivity of claw-free incomparability graphs, Hamel and her collaborators [\textit{A. M. Hamel} et al., Graphs Comb. 35, No. 4, 815--825 (2019; Zbl 1416.05105)] studied the \(e\)-positivity of \((claw, H)\)-free graphs, where \(H\) is a four-vertex graph. In this paper we establish the \(e\)-positivity of generalized pyramid graphs and \(2K_2\)-free unit interval graphs, which are two important families of \((claw, 2K_2)\)-free graphs. Hence we affirmatively solve one problem proposed by Hamel et al. [loc. cit.], and another problem considered by \textit{A. M. Foley} et al. [Electron. J. Comb. 26, No. 3, Research Paper P3.51, 20 p. (2019; Zbl 1420.05177)].
scientific article; zbMATH DE number 7361298
scientific article