Angelika Hellwig

Sufficient conditions for λ′-optimality in graphs of diameter 2

Abstract For a connected graph G the restricted edge-connectivity λ′(G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in G−S. We call a graph G λ′-optimal, if λ′(G)=ξ(G), where ξ(G) is the minimum edge-degree in G. In 1999, Wang and Li (J. Shanghai Jiaotong Univ. 33(6) (1999) 646) gave a sufficient condition for λ′-optimality in graphs of diameter 2. In this paper, we weaken this condition, and we present some related results. Different examples will show that the results are best possible and independent of each other and of the result of Wang and Li.

Note on the connectivity of line graphs

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G). A subset S of E(G) is called a restricted edge-cut if G - S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ2(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ2(G) ≥ λ(G) ≥ κ(G). In 1969, Chartrand and Stewart have shown that κ(L(G)) ≥ λ(G), if λ(G) ≥ 2. where L(G) denotes the line graph of G. In the present paper we show that κ(L(G)) = λ2(G), if |V (G)| ≥ 4 and G is not a star, which improves the result of Chartrand and Stewart. As a direct consequence of this identity, we obtain the known inequality λ2(G) ≤ ξ(G) by Esfahanian and Hakimi, where ξ(G) is the minimum edge degree.

The connectivity of a graph and its complement

Let G be a graph with minimum degree @d(G), edge-connectivity @l(G), vertex-connectivity @k(G), and let G@? be the complement of G. In this article we prove that either @l(G)=@d(G) or @l(G@?)=@d(G@?). In addition, we present the Nordhaus-Gaddum type result @k(G)+@k(G@?)>=min{@d(G),@d(G@?)}+1. A family of examples will show that this inequality is best possible.

Lower bounds on the vertex-connectivity of digraphs and graphs

Since interconnection networks are often modeled by graphs or digraphs, the connectivity of a (di-)graph is an important measurement for fault tolerance of networks.Let G be a graph of order n, minimum degree δ, and vertex-connectivity k. If G is not the complete graph, then Chartrand and Harary [G. Chartrand, F. Harary, Graphs with prescribed connectivities, in: P. Erdos, G. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 61-63] proved in 1968 that k≥2δ+2-n.In 1993, Topp and Volkmann [J. Topp, L. Volkmann, Sufficient conditions for equality of connectivity and minimum degree of a graph, J. Graph Theory 17 (1993) 695-700] proved the following analog bound for bipartite graphs. If G is a bipartite graph with k<δ then k≥4δ-n.In this paper we present some generalizations and extensions of these inequalities for digraphs as well as for graphs.

On the connectivity of diamond-free graphs

Let G be a graph of order n(G), minimum degree Δ(G) and connectivity κ(G). Chartrand and Harary [Graphs with prescribed connectivities, in: P. Erdos, G. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 61-63] gave the following lower bound on the connectivity κ(G)≥2Δ(G)+2-n(G).

Neighborhood and degree conditions for super-edge-connected bipartite digraphs

A graph or digraph D is called super-λ, if every minimum edge cut consists of edges incident to or from a vertex of minimum degree, where λ is the edge-connectivity of D. Clearly, if D is super-λ, then λ = δ, where δ is the minimum degree of D. In this paper neighborhood, degree sequence, and degree conditions for bipartite graphs and digraphs to be super-λ are presented. In particular, the neighborhood condition generalizes the following result by Fiol [7]: If D is a bipartite digraph of order n and minimum degree δ ≥ max{3, ⌈(n + 3)/4⌉}, then D is super-λ.