A note on triangle-free graphs

A note on the NP-completeness of the precoloring extension coloring problem in triangle free planar graphs

The precoloring extension coloring problem consists in deciding, given a positive integer k, a graph G = (V,E) and k pairwise disjoint subsets V0, . . . , Vk−1 of V , if there exists a (vertex) coloring S = (S0, . . . , Sk−1) of G such that Vi ⊆ Si, for all i = 0, . . . , k − 1. In this note, we show that the precoloring extension coloring problem is NP-complete in triangle free planar graphs w...

Rooted induced trees in triangle-free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. Further, for a vertex v ∈ V (G), let t(G, v) denote the maximum number of vertices in an induced subgraph of G that is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle-free graphs G (and vertices v ∈ V (G)...

The spectral radius of triangle-free graphs

In this note, we present two lower bounds for the spectral radius of the Laplacian matrices of triangle-free graphs. One is in terms of the numbers of edges and vertices of graphs, and the other is in terms of degrees and average 2-degrees of vertices. We also obtain some other related results.

Another Infinite Sequence of Dense Triangle-Free Graphs

The core is the unique homorphically minimal subgraph of a graph A triangle free graph with minimum degree n is called dense It was observed by many authors that dense triangle free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms One in nite sequence of cores of dense maximal triangle free graphs was...

On uniquely Hamiltonian claw-free and triangle-free graphs

A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can b...

A Note on Triangle-Free Graphs