On the Structure of Dense Triangle-Free Graphs

Abstract

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ[ges ]n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every triangle-free graph with δ>n/3 is either homomorphic with C5 or contains the Möbius ladder. A major tool is the observation that every triangle-free graph with δ[ges ]n/3 has a unique maximal triangle-free supergraph.

Graphs Whose Circular Chromatic Number Equals the Chromatic Number

“…Concerning graphs with circular chromatic numbers at least 3, there are some interesting characterizations of such graphs which follow from a recent result of Brandt [3] and an earlier result of Pach [11]. Both Brandt and Pach were aiming at some other problems.…”

The minimum number of vertices for a triangle-free graph with χl(G)=4 is 11

“…The following special class of maximal triangle-free graphs plays an important role in the investigation on dense triangle-free graphs, see [1]. For an integer k ∈ N , the graph F k has V (F k ) = {v 1 , v 2 , .…”

Subgraphs in vertex neighborhoods of Kr‐free graphs

“…In 1995, this result was rediscovered by Brouwer [5], who gave a considerably simpler proof. While the original characterization needs some introduction, which will be given later, the second author [4] observed that there are two forbidden subgraph formulations characterizing these graphs. A triangle-free graph has the property that every independent set is contained in the neighborhood of a vertex if and only if it is maximal triangle-free and contains no 6-cycle as an induced subgraph, or, equivalently, no vertex-deleted Petersen graph as a (not necessarily induced) subgraph.…”

“…A triangle-free graph has the property that every independent set is contained in the neighborhood of a vertex if and only if it is maximal triangle-free and contains no 6-cycle as an induced subgraph, or, equivalently, no vertex-deleted Petersen graph as a (not necessarily induced) subgraph. The forbidden subgraph characterizations turned out to be helpful for the application of Pach's result to problems on triangle-free graphs with large minimum degree [4].…”