arXiv subjects

Lixing Yi

Publications and source records attributed to Lixing Yi.

3 recordsLinked to original sources

arXiv2026

An improved upper bound for Tuza's conjecture via 2-colorable triangle families

Tuza's conjecture states that for any graph $G$, the minimum size of a triangle transversal $\tau(G)$ is at most twice the maximum size of a set of edge-disjoint triangles $\nu(G)$. In this note, we prove $\tau(G) \leq \frac{63}{22}\nu(G)$, improving the previous bound $\tau(G) \leq \frac{66}{23}\nu(G)$ established by Haxell in 1999. The key observation is that for a "2-colorable" family of triangles $\mathcal{F}$, where each triangle has two blue edges and one red edge, we can obtain $\tau(\mathcal{F})\leq (1+\sqrt{3})\nu(\mathcal{F})$.

math.CO↗

arXiv2025

Cycle systems, coparking functions, and h-vectors of matroids

The h-vector of a matroid M is an important invariant related to the independence complex of M and can also be recovered from an evaluation of its Tutte polynomial. A well-known conjecture of Stanley posits that the h-vector of a matroid is a pure O-sequence, meaning that it can be obtained by counting faces of a pure multicomplex. Merino has established Stanley's conjecture for the case of cographic matroids via chip-firing on graphs and the concept of a G-parking function. Inspired by these constructions, we introduce the notion of a cycle system for a matroid M -- a family of cycles (unions of circuits) of M with overlap properties that mimic cut-sets in a graph. A choice of cycle system on M defines a collection of integer sequences that we call coparking functions. We show that for any cycle system on M, the set of coparking functions is in bijection with the set of bases of M. We show that maximal coparking functions all have the same degree, and that cycle systems behave well under deletion and contraction. This leads to a proof of Stanley's conjecture for the case of matroids that admit cycle systems, which include, for instance, graphic matroids of cones as well as K33-free graphs.

math.CO↗

arXiv2025

Enumerating submonoids of finite commutative monoids

Given a finite commutative monoid $M$, we show that submonoids of $M\times [n]$ - where $[n] = \{0,1,\ldots,n\}$ is equipped with the max operation $\vee$ - may be enumerated via the transfer matrix method. When $M$ is also idempotent, we show that there are finitely many integers $\lambda$ and rational numbers $b_\lambda$ (only depending on $M$) such that the number of submonoids of $M\times [n]$ is $\sum_\lambda b_\lambda\lambda^n$. This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.

math.CO↗