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A multigraph is exactly k-edge-connected if there are exactly k edge-disjoint paths between any pair of vertices. We characterize the class of exactly 3-edge-connected graphs, giving a synthesis involving two operations by which every exactly 3-edge-connected multigraph can be generated. Slightly modified syntheses give the planar exactly 3-edge-connected graphs and the exactly 3-edge-connected graphs with the fewest possible edges.
A finite graph $G$ is said to be {em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $gin G$ of the graph, where $G$ is a group of automorphism
Halin showed that every edge minimal, k-vertex connected graph has a vertex of degree k. In this note, we prove the analogue to Halins theorem for edge-minimal, k-edge-connected graphs. We show there are two vertices of degree k in every edge-minimal, k-edge-connected graph.
This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automo
In 2006, Barat and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition into copies o
For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set correspondin