On embedding triangle-free graphs in unit spheres

On Generating Triangle-Free Graphs

We show that the problem to decide whether a graph can be made triangle-free with at most k edge deletions remains NP-complete even when restricted to planar graphs of maximum degree seven. In addition, we provide polynomial-time data reduction rules for this problem and obtain problem kernels consisting of 6k vertices for general graphs and 11k/3 vertices for planar graphs.

On Reed's Conjecture in Triangle-Free Graphs

The problem of finding the chromatic number of a graph G, i.e. the minimum number of colors needed to assign distinct colors to adjacent nodes of G, is NP-complete. However, there are some known bounds including the trivial lower bound χ(G) ≥ ω(G) and the upper bound provided by Brooks [2] χ(G) ≤ ∆(G) + 1, where χ(G) denotes the chromatic number, ∆(G) the maximum degree and ω(G) the number of n...

Measuring triangle-free graphs

Graphs with Eulerian unit spheres

d-spheres are defined graph theoretically and inductively as the empty graph in dimension d = −1 and d-dimensional graphs for which all unit spheres S(x) are (d−1)-spheres and such that for d ≥ 0 the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which can be colored with d + 1 colors. They are Eulerian graphs in the classical sense and for d ≥ ...

A note on triangle-free graphs

We show that if G is a simple triangle-free graph with n ≥ 3 vertices, without a perfect matching, and having a minimum degree at least n−1 2 , then G is isomorphic either to C5 or to Kn−1 2 , n+1 2 .