Homomorphisms into loop-threshold graphs

Jonathan Cutler, Nicholas Kass

Research output: Contribution to journalArticlepeer-review


Many problems in extremal graph theory correspond to questions involving ho-momorphisms 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 Hind, the graph on two adjacent vertices, one of which is looped, each homomorphism from G to Hind cor-responds to an independent set in G. It follows from the Kruskal-Katona theorem that the number of homomorphisms to Hind 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 Hind is a loop-threshold graph. We survey known results for maximizing the number of ho-momorphisms into small loop-threshold image graphs. The only extremal homo-morphism problem with a loop-threshold image graph on at most three vertices not yet solved is Hind ∪ E1, 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 ℓ(m), the number of vertices in the lex component of the extremal graph with m edges and at least m + 1 vertices.

Original languageEnglish
Article numberP2.38
Pages (from-to)1-18
Number of pages18
JournalElectronic Journal of Combinatorics
Issue number2
StatePublished - 2020


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