当前位置: X-MOL 学术arXiv.cs.DM › 论文详情
Multi-transversals for Triangles and the Tuza's Conjecture
arXiv - CS - Discrete Mathematics Pub Date : 2020-01-01 , DOI: arxiv-2001.00257
Parinya Chalermsook; Samir Khuller; Pattara Sukprasert; Sumedha Uniyal

In this paper, we study a primal and dual relationship about triangles: For any graph $G$, let $\nu(G)$ be the maximum number of edge-disjoint triangles in $G$, and $\tau(G)$ be the minimum subset $F$ of edges such that $G \setminus F$ is triangle-free. It is easy to see that $\nu(G) \leq \tau(G) \leq 3 \nu(G)$, and in fact, this rather obvious inequality holds for a much more general primal-dual relation between $k$-hyper matching and covering in hypergraphs. Tuza conjectured in $1981$ that $\tau(G) \leq 2 \nu(G)$, and this question has received attention from various groups of researchers in discrete mathematics, settling various special cases such as planar graphs and generalized to bounded maximum average degree graphs, some cases of minor-free graphs, and very dense graphs. Despite these efforts, the conjecture in general graphs has remained wide open for almost four decades. In this paper, we provide a proof of a non-trivial consequence of the conjecture; that is, for every $k \geq 2$, there exist a (multi)-set $F \subseteq E(G): |F| \leq 2k \nu(G)$ such that each triangle in $G$ overlaps at least $k$ elements in $F$. Our result can be seen as a strengthened statement of Krivelevich's result on the fractional version of Tuza's conjecture (and we give some examples illustrating this.) The main technical ingredient of our result is a charging argument, that locally identifies edges in $F$ based on a local view of the packing solution. This idea might be useful in further studying the primal-dual relations in general and the Tuza's conjecture in particular.
更新日期:2020-01-04

 

全部期刊列表>>
化学/材料学中国作者研究精选
Springer Nature 2019高下载量文章和章节
《科学报告》最新环境科学研究
ACS材料视界
自然科研论文编辑服务
中南大学国家杰青杨华明
剑桥大学-
中国科学院大学化学科学学院
材料化学和生物传感方向博士后招聘
课题组网站
X-MOL
北京大学分子工程苏南研究院
华东师范大学分子机器及功能材料
中山大学化学工程与技术学院
试剂库存
天合科研
down
wechat
bug