Disjoint subgraphs of large maximum degree

Disjoint subgraphs of large maximum degree

Large induced subgraphs with equated maximum degree

For a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f2(G) ≥ √ n(1− o(1)), where n is the number of vertices of G. It is conjectured that fk(G) = Θ( √ n) for every fixed k. We prove this for k = 2,...

Sizes of graphs with induced subgraphs of large maximum degree

Graphs with n + k vertices in which every set of n +j vertices induce a subgraph of maximum degree at least n are considered. For j = 1 and for k fairly small compared to n, we determine the minimum number of edges in such graphs. In investigating the size Ramsey number of a star KI,, versus a triangle K3, Erdiis [ 1,2] conjectured that for n 3 3 any graph with no more than (2”:‘) (‘;) 1 edges ...

Vertex-disjoint subgraphs with high degree sums

For a graph G, we denote by σ2(G) the minimum degree sum of two non-adjacent vertices if G is non-complete; otherwise, σ2(G) = +∞. In this paper, we give the following two results; (i) If s1 and s2 are integers with s1, s2 ≥ 2 and if G is a non-complete graph with σ2(G) ≥ 2(s1 + s2 + 1) − 1, then G contains two vertexdisjoint subgraphs H1 and H2 such that each Hi is a graph of order at least si...

Approximating Bounded Degree Maximum Spanning Subgraphs∗

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...