zbMATH — the first resource for mathematics

Every 4k-edge-connected graph is weakly 3k-linked. (English) Zbl 0708.05036
A graph is weakly k-linked, if for every k pairs of vertices \((s_ i,t_ i)\) there exist k edge-disjoint paths \(P_ i\) such that \(P_ i\) joins \(s_ i\) and \(t_ i\) (1\(\leq i\leq k)\). Let be g(k) the minimum of the edge connectivity numbers such that for each graph G which is g(k) edge connected holds: G is weakly k-linked. C. Thomassen [Eur. J. Comb. 1, 371-378 (1980; Zbl 0457.05044)] conjectured that \(g(2k+1)=g(2k)=2k+1\) (k\(\geq 1)\). Here the author proves g(3k)\(\leq 4k\) and \(g(3k+1)\leq g(3k+2)\leq 4k+2\) (k\(\geq 2)\).
Reviewer: M.Hager

05C40 Connectivity
05C38 Paths and cycles
weakly k-linked
Full Text: DOI
[1] Hirata, T., Kubota, K., Saito, O.: A sufficient condition for a graph to be weaklyk-linked. J. Comb. Theory (B)36, 85–94 (1984) · Zbl 0535.05042 · doi:10.1016/0095-8956(84)90015-7
[2] Mader, W.: A reduction method for edge-connectivity in graphs. Ann. Discrete Math.3, 145–164 (1978) · Zbl 0389.05042 · doi:10.1016/S0167-5060(08)70504-1
[3] Mader, W.: Paths in graphs, reducing the edge-connectivity only by two. Graphs and Combinatorics1, 81–89 (1985) · Zbl 0579.05037 · doi:10.1007/BF02582931
[4] Okamura, H.: Multicommodity flows in graphs II. Japan. J. Math.10, 99–116 (1984) · Zbl 0589.90025
[5] Okamura, H.: Paths and edge-connectivity in graphs. J. Comb. Theory (B)37, 151–172 (1984) · Zbl 0548.05039 · doi:10.1016/0095-8956(84)90069-8
[6] Okamura, H.: Paths and edge-connectivity in graphs II. In: Number theory and Combinatorics (Tokyo, Okayama, Kyoto 1984), pp. 337–352. Singapore: World Sci. Publishing 1985 · Zbl 0548.05039
[7] Okamura, H.: Paths and edge-connectivity in graphs III. Six-terminalk paths. Graphs and Combinatorics3, 159–189 (1987) · Zbl 0643.05047 · doi:10.1007/BF01788539
[8] Okamura, H.: Paths ink-edge-connected graphs. J. Comb. Theory (B)45, 345–355 (1988) · doi:10.1016/0095-8956(88)90077-9
[9] Thomassen, C.: 2-linked graphs. European J. Comb.1, 371–378 (1980) · Zbl 0457.05044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.