The maximum number of complete subgraphs in a graph with given maximum degree

The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and ∆(G) ≤ r, which has the most complete subgraphs of size t, for t ≥ 3. The conjectured extremal graph is aKr+1 ∪Kb, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan, Loh, and Sudakov proved the conjecture wh...

The maximum number of complete subgraphs in a graph with given maximum degree

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most (2 − 1) by the Kahn-Zhao theorem [7, 12]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for n ≥ 2d, the number of independent sets in a graph with δ(G) ≥ ...

The Maximum Number of Edges in a Strongly Multiplicative Graph

On the Maximum Number of Dominating Classes in Graph Coloring

In this paper we investigate the dominating- -color number،  of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and H. This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].  

The Ramsey number of a graph with bounded maximum degree

The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G. A conjecture of P. Erdos and S. Burr is settled in the afftrmative by proving that for each...