Abstract

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and edges into small image graphs. For the image graph $H_\text{ind}$, the graph on two adjacent vertices, one of which is looped, each homomorphism from $G$ to $H_\text{ind}$ corresponds to an independent set in $G$. It follows from the Kruskal-Katona theorem that the number of homomorphisms to $H_\text{ind}$ is maximized by the lex graph, whose edges form an initial segment of the lex order. 
 A loop-threshold graph is a graph built recursively from a single vertex, which may be looped or unlooped, by successively adding either a looped dominating vertex or an unlooped isolated vertex at each stage. Thus, the graph $H_\text{ind}$ is a loop-threshold graph. We survey known results for maximizing the number of homomorphisms into small loop-threshold image graphs. The only extremal homomorphism problem with a loop-threshold image graph on at most three vertices not yet solved is $H_\text{ind}\cup E_1$, where extremal graphs are the union of a lex graph and an empty graph. The only question that remains is the size of the lex component of the extremal graph. While we cannot give an exact answer for every number of vertices and edges, we establish the significance of and give bounds on $\ell(m)$, the number of vertices in the lex component of the extremal graph with $m$ edges and at least $m+1$ vertices.

Highlights

  • Many problems in classical extremal graph theory can be stated in terms of graph homomorphisms

  • If we take the image graph to be Hind, a path on two vertices with one vertex looped, elements of hom(G, Hind) correspond to independent sets in G. This is because vertices mapped to the unlooped vertex of Hind form an independent set and there are no other restrictions on the map

  • Threshold graphs play a key role in the investigation of extremal problems related to graph homomorphisms

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Summary

Introduction

Many problems in classical extremal graph theory can be stated in terms of graph homomorphisms. If we take the image graph to be Hind, a path on two vertices with one vertex looped (see Figure 1), elements of hom(G, Hind) correspond to independent sets in G. The solution to the problem of maximizing the number of independent sets over G(n, m) follows from the Kruskal-Katona [16, 15] theorem. This is because independent sets in a graph G correspond exactly to those sets that are not in the upper shadow of G, where G is thought of as a set system on the vertex set.

Threshold and loop-threshold graphs
The remaining case
An extremal question
Future directions
Full Text
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