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Mod (2p+1)-orientations in line graphs

Authors:	Hong-Jian Lai Hao Li Ping Li Yanting Liang Senmei Yao

Affiliation:	1. Department of Pediatric Surgery, Children''s Hospital of Fudan University, Shanghai Key Laboratory of Birth Defect, Shanghai 201102, China;2. Department of General Surgery, Jiangxi Provincial Children''s Hospital, Nanchang, Jiangxi Province, 330006, China;3. Medical Scientific Liaison Asian Pacific, Abbott Diagnostics Division, Abbott Laboratories, Shanghai 200032, China;1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China;2. Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA;1. Department of Management Sciences, College of Business, City University of Hong Kong, Hong Kong;2. School of Economics and Management, Wuhan University, Wuhan, 430072, PR China

Abstract:	Jaeger in 1984 conjectured that every $(4 p)$ -edge-connected graph has a mod $(2 p + 1)$ -orientation. It has also been conjectured that every $(4 p + 1)$ -edge-connected graph is mod $(2 p + 1)$ -contractible. In [Z.-H. Chen, H.-J. Lai, H. Lai, Nowhere zero flows in line graphs, Discrete Math. 230 (2001) 133–141], it has been proved that if G has a nowhere-zero 3-flow and the minimum degree of G is at least 4, then $L (G)$ also has a nowhere-zero 3-flow. In this paper, we prove that the above conjectures on line graphs would imply the truth of the conjectures in general, and we also prove that if G has a mod $(2 p + 1)$ -orientation and $δ (G) ? 4 p$ , then $L (G)$ also has a mod $(2 p + 1)$ -orientation, which extends a result in Chen et al. (2001) [2].

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