We consider the question: “What is the smallest degree that can be achieved for a plane spanner of a Euclidean graph E?” The best known bound on the degree is 14. We show that E always contains a plane spanner of maximum degree 6 and stretch factor 6. This spanner can be constructed efficiently in linear time given the Triangular Distance Delaunay triangulation introduced by Chew.

The bounded degree maximum spanning subgraph problem arising from wireless mesh networks is studied here. Given a connected graph G and a positive integer d ≥ 2, the problem aims to find a maximum spanning subgraph H of G with the constraint: for every vertex v of G, the degree of v in H, dH(v), is less than or equal to d. Here, a spanning subgraph is a connected subgraph which contains all the...

Let S(H) be the minimum degree of the graph H. We prove that a graph H of order n with S(H) ^ (2n —1)/3 contains any graph G of order at most n and maximum degree A(G) < 2 as a subgraph, and this bound is best possible. Furthermore, this result settles the case A(G) = 2 of the well-known conjecture of Bollobas, Catlin and Eldridge on packing two graphs with given maximum degree.

For a graph Gand an integer k, denote by Vk the set {u E V(G) I d(u) 2 k}. Veldman proved that if G is a 2-connected graph of order n with n 5 3k 2 and IVkl 5 k, then G has a cycle containing all vertices of Vk. It is shown that the upper bound k on IVkl is close to best possible in general. For the special case k = A(G), it is conjectured that the condition lVkl I k can be omitted. Using a var...