Eulerian subgraphs and Hamilton-connected line graphs

Authors:
Dengxin Li;Hong-Jian Lai;Mingquan Zhan
Affiliations:
Department of Mathematics, Chongqing Technology and Business University, Chongqing, P. R. China;Department of Mathematics, West Virginia University, Morgantown, WV;Department of Mathematics, Millersville University, Millersville, PA
Venue:
Discrete Applied Mathematics
Year:
2005

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Abstract

Let C(l,k) denote a class of 2-edge-connected graphs of order n such that a graph G ∈ C(l,k) if and only if for every edge cut S ⊆ E(G) with |S| ≤ 3, each component of G-S has order at least (n-k)/l. We prove the following: (1) If G ∈ C(6,0), then G is supereulerian if and only if G cannot be contracted to K2,3, K2,5 or K2,3(e), where e ∈ E(K2,3) and K2,3(e) stands for a graph obtained from K2,3 by replacing e by a path of length 2. (2) If G ∈ C(6, 0) and n≥7, then L(G) is Hamilton-connected if and only if κ(L(G))≥3. Former results by Catlin and Li, and by Broersma and Xiong are extended.