Bound on shortest cycle covers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916552070135808 |
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| author | Song, Deping Zhu, Xuding |
| author_facet | Song, Deping Zhu, Xuding |
| contents | Assume $G$ is a bridgeless graph. A cycle cover of $G$ is a collection of cycles of $G$ such that each edge of $G$ is contained in at least one of the cycles. The length of a cycle cover of $G$ is the sum of the lengths of the cycles in the cover. The minimum length of a cycle cover of $G$ is denoted by $cc(G)$. It was proved independently by Alon and Tarsi and by Bermond, Jackson, and Jaeger that $cc(G)\le \frac{5}{3}m$ for every bridgeless graph $G$ with $m$ edges. This remained the best-known upper bound for $cc(G)$ for 40 years. In this paper, we prove that if $G$ is a bridgeless graph with $m$ edges and $n_2$ vertices of degree $2$, then $cc(G) < \frac{29}{18}m+ \frac 1{18}n_2$. As a consequence, we show that $cc(G) \le \frac 53 m - \frac 1{42} \log m$. The upper bound $ cc(G) < \frac{29}{18}m \approx 1.6111 m$ for bridgeless graphs $G$ of minimum degree at least 3 improves the previous known upper bound $1.6258m$. A key lemma used in the proof confirms Fan's conjecture that if $C$ is a circuit of $G$ and $G/C$ admits a nowhere zero 4-flow, then $G$ admits a 4-flow $f$ such that $E(G)-E(C)\subseteq \text{supp} (f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bound on shortest cycle covers Song, Deping Zhu, Xuding Combinatorics Assume $G$ is a bridgeless graph. A cycle cover of $G$ is a collection of cycles of $G$ such that each edge of $G$ is contained in at least one of the cycles. The length of a cycle cover of $G$ is the sum of the lengths of the cycles in the cover. The minimum length of a cycle cover of $G$ is denoted by $cc(G)$. It was proved independently by Alon and Tarsi and by Bermond, Jackson, and Jaeger that $cc(G)\le \frac{5}{3}m$ for every bridgeless graph $G$ with $m$ edges. This remained the best-known upper bound for $cc(G)$ for 40 years. In this paper, we prove that if $G$ is a bridgeless graph with $m$ edges and $n_2$ vertices of degree $2$, then $cc(G) < \frac{29}{18}m+ \frac 1{18}n_2$. As a consequence, we show that $cc(G) \le \frac 53 m - \frac 1{42} \log m$. The upper bound $ cc(G) < \frac{29}{18}m \approx 1.6111 m$ for bridgeless graphs $G$ of minimum degree at least 3 improves the previous known upper bound $1.6258m$. A key lemma used in the proof confirms Fan's conjecture that if $C$ is a circuit of $G$ and $G/C$ admits a nowhere zero 4-flow, then $G$ admits a 4-flow $f$ such that $E(G)-E(C)\subseteq \text{supp} (f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$. |
| title | Bound on shortest cycle covers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.20874 |