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- Title
Supereulerian regular matroids without small cocircuits.
- Authors
Huo, Bofeng; Du, Qingsong; Li, Ping; Wu, Yang; Yin, Jun; Lai, Hong‐Jian
- Abstract
A cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large n $n$, every 2‐edge‐connected simple graph G $G$ with n=∣V(G)∣ $n=| V(G)| $ and minimum degree δ(G)≥n5 $\delta (G)\ge \frac{n}{5}$ is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765–1776], it is shown that for any connected simple regular matroid M $M$, if every cocircuit D $D$ of M $M$ satisfies ∣D∣≥maxr(M)−55,6 $| D| \ge \max \left\{\frac{r(M)-5}{5},6\right\}$, then M $M$ is supereulerian. We prove the following. (i) Let M $M$ be a connected simple regular matroid. If every cocircuit D $D$ of M $M$ satisfies ∣D∣≥maxr(M)+110,9 $| D| \ge \max \left\{\frac{r(M)+1}{10},9\right\}$, then M $M$ is supereulerian. (ii) For any real number c $c$ with 0<c<1 $0\lt c\lt 1$, there exists an integer f(c) $f(c)$ such that if every cocircuit D $D$ of a connected simple cographic matroid M $M$ satisfies ∣D∣≥max{c(r(M)+1),f(c)} $| D| \ge \max \{c(r(M)+1),f(c)\}$, then M $M$ is supereulerian.
- Subjects
MATROIDS; GRAPH theory; REAL numbers; COMBINATORICS
- Publication
Journal of Graph Theory, 2023, Vol 102, Issue 1, p107
- ISSN
0364-9024
- Publication type
Article
- DOI
10.1002/jgt.22860