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...