A degree sum condition for the existence of a quasi 5-contractible edge in a quasi 5-connected graph

TL;DR

The paper addresses the existence of quasi $5$-contractible edges in quasi $5$-connected graphs. It develops a fragment/atom framework and analyzes edge-contraction effects to prove two main results: (1) every $5$-connected graph contains a quasi $5$-contractible edge, and (2) if a quasi $5$-connected graph satisfies the degree-sum condition $d_G(u)+d_G(v)\ge 9$ for every pair of vertices at distance $1$ or $2$, then a quasi $5$-contractible edge exists. The methods combine contraction-critical graph analysis with detailed fragment, atom, and neighborhood-type classifications to rule out the possibility of contraction-critical counterexamples. The results strengthen Kriesell’s degree-sum condition for $k$-connected graphs and extend contraction-closure properties to the quasi $5$-connected setting, with implications for the robustness of high-connectivity under edge contractions.

Abstract

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we prove that a quasi 5-connected graph possesses a quasi 5-contractible edge, if the degree sum of any two vertices with distance at most two is at least 9. This result strengthens a theorem proved by Kriesell when $k=4$ (M. Kriesell, A degree sum condition for the existence of a contractible edge in a $k$-connected graph, J. Combin. Theory Ser. B 82(2001)81-101).

Figures (2)

• Figure 1: A black vertex indicates that the vertex has reached its maximum degree.
• Figure 2: $K_{2}\cup 2K_{1}$, $P_{3}\cup K_{1}$, $2K_{2}$, $P_{4}$, $C_{4}$, $K_{1,3}$

Theorems & Definitions (24)

• Theorem A
• Theorem B
• Theorem C
• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4