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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.

11), V(P*) = {u~, x, y, u2} for some x ~ X2 and y e X 1 . 23), there is (k + 3)-set Z with Z f ) ( T U {x,y})-(vl, u2, x}. e(X1 tq Z) >_ k + 4, and e(X1 tq Z) > k + 4 since X1 and X2 are minimal (k + 3)-sets, contrary to Lemma 2. 25), V(G) - T is an independent set. 22). 6). 11). This completes the proof of Propositions A, B and C and Theorem 1. [] Acknowledgment. The author would like to thank the referee for his helpful comments. References 1. : On a theorem of Mader. Discrete Mathematics 101, 49-57 (1992) 2.

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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2-reducible cycles containing three consecutive edges in (2k + 1)-edge-connected graphs by Okamura H.


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