The Size of a Graph Without Topological Complete Subgraphs

Abstract

Abstract. In this note we show a new upperbound for the function ex(n; T K p ), i.e., the maximum number of edges of a graph of order n not containing a subgraph homeomorphic to the complete graph of order p. Further, for 2n+5 3≤ p < n we provide exact values for this function.

Extremal Graphs without Topological Complete Subgraphs

“…The next results show different conditions to guarantee that a graph belongs to the family described above (see [2]). …”

“…Lemma 2.1 (see [2] [2]). Let n and q be two positive integers, with q < n. If H is a graph with n vertices and 2q edges, then…”

“…Lemma 5.2 (see [2]). Let k be a nonnegative integer and H be a graph with maximum degree 2 and at least 3k + 1 vertices of maximum degree.…”

“…Exact values for this function are known only in some cases, as can be seen in Table 1.1. ≤ p < n n 2 − (2n − 2p + 1) [2] The aim of this work is to characterize a family of extremal graphs EX(n; T K p ) for appropriate values of n and p, i.e., the set of graphs of order n, with ex(n; T K p ) edges and not containing any subgraph homeomorphic to K p . Actually, we characterize the family EX(n; T K p ) for ≤ p < n − 3.…”

An advance in infinite graph models for the analysis of transportation networks

“…In general, and not only in extremal graph theory, we can find many problems involving the relationship between the numbers of vertices and edges of a graph (see, e.g., Cera et al, 2000;, Yang et al, 2002Yousefi-Azaria et al, 2011), i.e., the average degree. In some cases, many of these problems could be posed for infinite graphs.…”