Tropicalizing the Graph Profile of Some Almost-Stars

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-04-22 DOI:10.1137/23m1594947

Maria Dascălu, Annie Raymond

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1351-1368, June 2024.
Abstract. Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs [math], the tropicalization of the graph profile of [math] essentially records all valid pure binomial inequalities involving graph homomorphism numbers for graphs in [math]. Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for the graph containing a single vertex as well as stars where one edge is subdivided. This allows pure binomial inequalities in homomorphism numbers (or densities) for these graphs to be verified through an explicit linear program where the number of variables is equal to the number of edges in the biggest graph involved.

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将一些近似恒星的图谱热带化

SIAM 离散数学杂志》第 38 卷第 2 期第 1351-1368 页，2024 年 6 月。摘要。极值组合学中的许多重要问题可以说是图同态数的多项式不等式证明，特别是许多问题要求证明纯二项式不等式。对于一个固定的图集合[math]，[math]图轮廓的热带化本质上记录了[math]中所有涉及图同态数的有效纯二项式不等式。基于布莱克曼和雷蒙德在 2022 年描述的想法和技术，我们计算了包含单个顶点的图以及一条边被细分的星图的图轮廓的热带化。这样，这些图的同态数（或密度）的纯二项式不等式就可以通过一个显式线性程序来验证，其中变量的数量等于所涉及的最大图中的边的数量。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.