On $α$-Critical Edges in König-Egerváry Graphs

Tuza's Conjecture for Graphs of Maximum Average Degree Less Than 7

Tuza's Conjecture states that if a graph $G$ does not contain more than $k$ edge-disjoint triangles, then some set of at most $2k$ edges meets all triangles of $G$. We prove Tuza's Conjecture for all graphs $G$ having no subgraph with average degree at least $7$. As a key tool in the proof, we introduce a notion of reducible sets for Tuza's Conjecture; these are substructures which cannot occur in a minimal counterexample to Tuza's Conjecture. We also introduce weak König--Egerváry graphs, a generalization of the well-studied König--Egerváry graphs.

On $α^{++}$-Stable Graphs

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals alpha(G) + mu(G), where mu(G) is the cardinality of a maximum matching in G. In this paper we characterize $α^{++}$-stable graphs, namely, the graphs whose stability numbers are invariant to adding any two edges from their complements. We show that a König-Egerváry graph is $α^{++}$-stable if and only if it has a perfect matching consisting of pendant edges and no …

On $α$-Square-Stable Graphs

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an $α$-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*G), where G*G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also show that G is an square-stable Koenig-Egervary graph if and only if it has a perfect match…

再论近似二部非 König-Egerváry 图

Almost Bipartite non-König-Egerváry Graphs Revisited

本文研究具有唯一奇圈的近似二部非 König-Egerváry 图，刻画其最大匹配、diadem、corona 与删点后成为 König-Egerváry 图之间的结构关系。

An Improved Approximation for Maximum $k$-Dependent Set on Bipartite Graphs

We present a $(1+\frac{k}{k+2})$-approximation algorithm for the Maximum $k$-dependent Set problem on bipartite graphs for any $k\ge1$. For a graph with $n$ vertices and $m$ edges, the algorithm runs in $O(k m \sqrt{n})$ time and improves upon the previously best-known approximation ratio of $1+\frac{k}{k+1}$ established by Kumar et al. [Theoretical Computer Science, 526: 90--96 (2014)]. Our proof also indicates that the algorithm retains its approximation ratio when applied to the (more general) class of König-Egerváry graphs.

连接湮灭数与独立数的不等式

Inequalities Connecting the Annihilation and Independence Numbers

本文研究了图的湮灭数与独立数之间的差异，针对树、二部图和König-Egerváry图证明了一系列紧上界不等式。

最大度至多为7的图的Tuza猜想

Tuza's conjecture for graphs of maximum degree at most seven

该研究证明最大度至多为7的图满足Tuza猜想，通过改进Puleo的可归约集框架，结合机器验证的局部证书完成极小7-正则反例的分析，明确了猜想常数的最优性。