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Additional resources for 2-reducible cycles containing three consecutive edges in (2k + 1)-edge-connected graphs

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Paths and edge-connectivity in graphs. J. Comb. Theory Ser. B 37, 151172 (1984) 6. : Paths in k-edge-connected graphs. J. Comb. Theory Ser. B 45, 345-355 (1988) 7. : Cycles containing three consecutive edges in 2k-edge-connected graphs, Topics in Combinatorics and Graph Theory (eds. R. Bodendiek and R. Henn), PhysieaVerlag Heidelberg (1991), 549-553 8. : 2-reducible cycles containing two specified edges in (2k + 1)-edgeconnected graphs, Contemporary Math.

In (A) and (B), if x e X1 vl, then e(x, X 2 ) > ~ + 1, and so IXxl = 2 since e(v~,X2)> 2, R 2 = (x) and e(v~) = k + 1, which contradicts e(v~)= k in (A). 11). In (T), we can choose P* such that IE(P*)N O(X1)I = 1. For otherwise, starting from Ux along P*, let z be the first vertex in X1, then X 2 ÷ v~ separates z from u2. Then there are disjoint D~, D2 c_ V(G) - (X2 + vl) such that V(G) = Dx O D 2 (3 X 2 (J {01 }, e(D1, D2) --- 0, z 6 D1 and u 2 ~ D 2. 8). Thus Xx = DIUD2U{vl} = {vt,v2,z}. 6) e(z, X 2 ) < ~ and e(z, v l) < ~, a contradiction.

Ann. Discrete Math. 3, 145-164 (1978) 4. : Paths in graphs, reducing the edge-connectivity only by two. Graphs and Comb. I, 81-89 (1985) 5. : Paths and edge-connectivity in graphs. J. Comb. Theory Ser. B 37, 151172 (1984) 6. : Paths in k-edge-connected graphs. J. Comb. Theory Ser. B 45, 345-355 (1988) 7. : Cycles containing three consecutive edges in 2k-edge-connected graphs, Topics in Combinatorics and Graph Theory (eds. R. Bodendiek and R. Henn), PhysieaVerlag Heidelberg (1991), 549-553 8. : 2-reducible cycles containing two specified edges in (2k + 1)-edgeconnected graphs, Contemporary Math.

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