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Two notions of unit distance graphs
, 2014
"... A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite sub-set of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the fir ..."
Abstract
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A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite sub-set of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed d the number of faithful distance graphs in Rd on n labelled vertices is 2(1+o(1))dn log2 n, and give a short proof of the known fact that the number of distance graphs in Rd on n labelled vertices is 2(1−1/bd/2c+o(1))n2/2. We also study the behavior of several Ramsey-type quantities involving these graphs. In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in Rd.
Many T copies in H-free graphs
"... For two graphs T and H with no isolated vertices and for an integer n, let ex(n, T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T = K2 is a single edge is the main subject of extremal graph theory. In the present paper we inve ..."
Abstract
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For two graphs T and H with no isolated vertices and for an integer n, let ex(n, T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T = K2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) ex(n,K3, C5) ≤ (1 + o(1))
