Abstract
Motivated by Stanley and Stembridge's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators 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, Hoáng and Tuero, and another problem considered by Foley, Hoáng and Merkel.
Highlights
Given a finite simple graph G with vertex set V and edge set E, a proper coloring of G is a function κ from V to P = {1, 2, . . .} such that κ(u) = κ(v) whenever uv ∈ E
It is clear that XG is a homogeneous symmetric function of degree n, where n is the cardinality of V
A well known basis is composed of elementary symmetric functions which are indexed by integer partitions
Summary
Hoang and Tuero proved the e-positivity for F = paw and F = co-paw They showed that a (claw, F )-free the electronic journal of combinatorics 28(2) (2021), #P2.40 graph is not necessarily e-positive if F is a diamond, co-claw, K4, 4K1, 2K2 or C4. Hoang and Tuero showed that if a peculiar graph is (claw, co-diamond, 2K2)-free, it can be characterized as a generalized pyramid GP(r, s, t), as illustrated, where a, b, c are three pairwise nonadjacent vertices, the vertices of Sa,b (Sa,c or Sb,c) form a clique of size r
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