Critical graphs without triangles: An optimum density construction

Critical graphs without triangles: An optimum density construction

We construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k ≥ 4. For k ≥ 6 our constructions have > ( 1 4 −ε)n edges, which is asymptotically best possible by Turán’s theorem. We also demonstrate (nonconstructively) the existence of dense k-critical graphs avoiding all odd cycles of length ≤ l for any l and any k ≥ 4, again with a best possible density of > ( 1 ...

Planar 4-critical graphs with four triangles

By the Grünbaum-Aksenov Theorem (extending Grötzsch’s Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such graphs. This answers a question of Erdős from 1990.

Density Conditions For Triangles In Multipartite Graphs

We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erdős, we prove that, if the minimum degree of G is at least 17|V (G)|/20, the largest triangle-free subgraphs are precisely the largest bipartite subgraphs in G. We investigate in particular the case where G is a complete multipartite graph. We prove that a finite trip...

Triangles in contraction critical 5-connected graphs

A Construction of Graphs without Triangles Having Pre - Assigned Order and Chromatic Number

The chromatic number x(r) of a combinatorial graph r is the least cardinal number a such that the set of nodes of I’ can be divided into a subsets so that every edge of P joins nodes belonging to different subsets. It is known? that corresponding to every finite a there exists a finite graph Pa without triangles satisfying x(P,) = a. In [l], Theorem 2, we have extended this result to transflnit...