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