Displaying similar documents to “Decomposing infinite 2-connected graphs into 3-connected components.”

Rainbow Connection In Sparse Graphs

Arnfried Kemnitz, Jakub Przybyło, Ingo Schiermeyer, Mariusz Woźniak (2013)

Discussiones Mathematicae Graph Theory

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An edge-coloured connected graph G = (V,E) is called rainbow-connected if each pair of distinct vertices of G is connected by a path whose edges have distinct colours. The rainbow connection number of G, denoted by rc(G), is the minimum number of colours such that G is rainbow-connected. In this paper we prove that rc(G) ≤ k if |V (G)| = n and for all integers n and k with n − 6 ≤ k ≤ n − 3. We also show that this bound is tight.

Connected domatic number in planar graphs

Bert L. Hartnell, Douglas F. Rall (2001)

Czechoslovak Mathematical Journal

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A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G . The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V ( G ) . We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic...