A sufficient condition for a graph to be weakly k-linked

Paths and edge-connectivity in graphs III. Six-terminalk paths

“…(st, x), k), then (1.9) holds. (2) If there exists an f e g(s~, tt) and T -{s~, tt } ~ F(G -f, k), then (1.9) holds.…”

“…Assume that for some f~( a l ) and ge~ (2,3), L(al,ai: G -{f (4 and a~EX~ (i=1,2,3). Thus e ( X i ; G t ) > _ k -1 (i=2,3).…”

“…(1), (2) and (4)are given in [5, L 6.21 and (3)in [6, L 2.5]. (5) By comparing e(2) with e(4), e(1)with e(3), and e(1)with e(2) we have e(2, 3) = e (1,4), r(1,2,4) = r (3,2,4) and e(1,3)= e(2,4) + r (2,3,4). Thus c (1,4) [6, L2.7]).…”

“…Suppose that k >_ 3 is odd, G is elemental for T = {al,a2,a3,a4,a5} and k, (1) If A (1,2) --~, then for some i ---1,2 and j = 3, 4, e(i,j) > O. (2) For each x~R (1,2,3) and each i = 3, 4, G -x has a path P [1, i] such that T e F(G -x -E(P), k -2) and P passes through at most two elemental stars.…”

“…Set ~ := r(1,2,4)+ r (1,3,4)+ r (2,3,4). (2) e (1)+e (2) By comparing e(4) with e (1), and e (4) with (2.2) e(1, 3)= e(4, 3)and e(1,2)= e (4,2).…”