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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.
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
Given any graph $H$, a graph $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges of $G$ with $q$ colors yields a monochromatic subgraph isomorphic to $H$. Further, such a graph $G$ is said to be minimal $q$-Ramsey for $H$ if additiona
A graph $G$ is $k$-edge-Hamiltonian if any collection of vertex-disjoint paths with at most $k$ edges altogether belong to a Hamiltonian cycle in $G$. A graph $G$ is $k$-Hamiltonian if for all $Ssubseteq V(G)$ with $|S|le k$, the subgraph induced by
A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $ell leq k$ can be connected by $ell$ disjoint paths in the whole graph. We characterise the existence of
Let $G$ be a simple graph with maximum degree $Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Delta(G)lfloor |V(H)|/2 rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $Delta(G)>|V(G)|/3$ has chromatic index $Delta(G)$ i