Events
DMS Combinatorics Seminar |
| Time: Jan 21, 2026 (01:00 PM) |
| Location: 328 Parker Hall |
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Details: Speaker: Guantao Chen (Georgia State University) Title: Adjacent cubic vertices in a minimal brick
Abstract: A connected graph \(G\) is called a matching covered graph if \(E(G)\neq\emptyset\) and every edge of \(G\) lies in a perfect matching, that is, a matching covering all vertices of \(G\). A matching covered graph is said to be bicritical if, for any two distinct vertice \(x,y\in V(G)\), the graph \(G-x-y\) has a perfect matching. A brick is a 3-connected bicritical graph, and it is minimal if the deletion of any edge destroys this property. A well-known conjecture of Lov\'asz asserts that every minimal brick contains two adjacent cubic (degree-3) vertices. In this talk, we present progress toward Lovász’s conjecture and establish several partial results. In particular, we show that every minimal brick contains two cubic vertices at distance at most two. We also verify the conjecture for minimal bricks with average degree at least \(4.5\). As a consequence, we deduce that every minimal brick contains two adjacent vertices of degree at most five. |