An e-positive classification for complete multipartite graphs
Abstract
Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~G=K(3, 2β) 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~XG 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.
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BibTeX
@misc{sun2026epositive,
title = {{An $e$-positive classification for complete multipartite graphs}},
author = {Ariel Y. Sun and David G. L. Wang and Watson Z. Y. Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.19110},
eprint = {2607.19110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19110},
}
RIS
TY - UNPB
TI - An e-positive classification for complete multipartite graphs
AU - Ariel Y. Sun
AU - David G. L. Wang
AU - Watson Z. Y. Wang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.19110
UR - https://arxiv.org/abs/2607.19110
AB - Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~G=K_(3, 2^β) 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.
ER -