## Abstract

Let *G* be a bridgeless graph and denote by *cc*(*G*) the shortest length of a cycle cover of *G*. Let *V*_{2}(*G*) be the set of vertices of degree 2 in *G*. It is known that if *cc*(*G*)≤1.4|*E*(*G*)| for every bridgeless graph *G* with |*V*_{2}(*G*)|≤\(\frac{1}{10}\)|*E*(*G*)|, then the Cycle Double Cover Conjecture is true. The best known result *cc*(*G*)≤\(\frac{5}{3}\)|*E*(*G*)| (≈1.6667|*E*(*G*)|) was established over 30 years ago. Recently, it was proved that *cc*(*G*) ≤ \(\frac{44}{27}\)|*E*(*G*)| (≈ 1.6296|*E*(*G*)|) for loopless graphs with minimum degree at least 3. In this paper, we obtain results on integer 4-flows, which are used to find bounds for *cc*(*G*). We prove that if *G* has minimum degree at least 3 (loops being allowed), then *cc*(*G*)<1.6258|*E*(*G*)|. As a corollary, adding loops to vertices of degree 2, we obtain that *cc*(*G*)<1.6466|*E*(*G*)| for every bridgeless graph *G* with |*V*_{2}(*G*)|≤\(\frac{1}{30}\)|*E*(*G*)|.

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Research supported by NSFC Grant 11331003.

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Fan, G. Integer 4-Flows and Cycle Covers.
*Combinatorica* **37, **1097–1112 (2017). https://doi.org/10.1007/s00493-016-3379-9

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### Mathematics Subject Classification (2000)

- 05C70
- 05C38
- 05C15