An $e$-positive classification for complete multipartite graphs

Ariel Y. Sun; David G. L. Wang; Watson Z. Y. Wang

An $e$-positive classification for complete multipartite graphs

Combinatorics 2026-07-21 v1

Authors: Ariel Y. Sun , David G. L. Wang , Watson Z. Y. Wang

Abstract

Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~ $G=K_{(3,\,2^\beta)}$ are $e$ -positive. We resolve this question and, together with their classification, characterize all $e$ -positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$ -expansion of~ $X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$ -coefficients of arbitrary complete multipartite graphs from the elementary--monomial Cauchy identity. For the particular graph~ $G$ , this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.

Cite

@article{arxiv.2607.19110,
  title  = {An $e$-positive classification for complete multipartite graphs},
  author = {Ariel Y. Sun and David G. L. Wang and Watson Z. Y. Wang},
  journal= {arXiv preprint arXiv:2607.19110},
  year   = {2026}
}