PreprintCC BY 4.0arXiv
Edge ideals of gapfree graphs with matching number $3$
Abstract
We prove that if G is a gapfree graph with matching number $3$, then every power I(G)q, q≥2, of its edge ideal has linear quotients, and hence a linear resolution. Our approach combines graph reductions based on clique domination of gapfree graphs and matching number with operations preserving linear quotients. We reduce the problem to $26$ non-co-chordal graphs on at most seven vertices. A new preservation theorem for gapfree edge addition, together with known results on duplication and gapfree expansion, handles all but one of these graphs. For the remaining graph on seven vertices, we construct a linear quotients ordering for the square and extend it recursively to every higher power.
Related
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BibTeX
@misc{erey2026edge,
title = {{Edge ideals of gapfree graphs with matching number $3$}},
author = {Nursel Erey and Tài Huy Hà and Takayuki Hibi and Selvi Kara and Susan Morey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21866},
eprint = {2610.21866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21866},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Edge ideals of gapfree graphs with matching number $3$
AU - Nursel Erey
AU - Tài Huy Hà
AU - Takayuki Hibi
AU - Selvi Kara
AU - Susan Morey
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21866
UR - https://arxiv.org/abs/2610.21866
AB - We prove that if G is a gapfree graph with matching number $3$, then every power I(G)^q, q≥2, of its edge ideal has linear quotients, and hence a linear resolution. Our approach combines graph reductions based on clique domination of gapfree graphs and matching number with operations preserving linear quotients. We reduce the problem to $26$ non-co-chordal graphs on at most seven vertices. A new preservation theorem for gapfree edge addition, together with known results on duplication and gapfree expansion, handles all but one of these graphs. For the remaining graph on seven vertices, we construct a linear quotients ordering for the square and extend it recursively to every higher power.
ER -