Triangle-Free Graphs with Large Degree

Abstract

A graph G is called an H-type graph for some graph H if there is a mapping from V (G) to V (H) preserving edges. In this paper, we shall prove that: (1) every triangle-free graph G of order n with χ(G) 6 3 and δ(G) > n/3 is of F d-type for some d > 1, where F d is a certain d-regular triangle-free Hamiltonian Cayley graph of order 3d − 1, (2) every triangle-free graph G of order n with χ(G) > 4 and δ(G) > n/3 contains the Mycielski graph (see Figure 2) as a subgraph.

Dense graphs with small clique number

“…Thomassen [16] proved that the chromatic number of triangle-free graphs with ≥ cn with c > 1 homomorphic to the 5-cycle. This result was extended by Jin [12] and Chen et al [6], inter alia. Łuczak [14] provided a general result about the structure of triangle-free graphs with minimum degree more than n / 3, which was made more precise in [5]: Ł Theorem 1.4 ( uczak [14], Brandt and Thomassé [5]).…”

“…the existence of homomorphisms [6,11,12,14], the existence of similarity sets [8], and bounds on the binding number [15,17]. On the other hand, K r -free graphs with large minimum degree are not as studied.…”

Sparse halves in dense triangle-free graphs

“…Corollary 2.6 (Chen, Jin, Koh [2], [7]). Every triangle-free graph G of order n with δ(G) ≥ 5 14 n is of F 5 -type.…”

On the local density problem for graphs of given odd-girth

“…It is easy to check that Andrásfai graphs are triangle-free and balanced blow-ups of these graphs play a prominent rôle in connection with extremal problems for triangle-free graphs (see, e.g., [1,2,6,7]). Our first result validates Conjecture 1.1 (stated in the contrapositive) for graphs homomorphic to some Andrásfai graph.…”

“…Owing to the work of Chen, Jin, and Koh [2], which asserts that every triangle-free 3-chromatic n-vertex graph G with minimum degree δpGq ą n{3 is homomorphic to some…”