Triply Existentially Complete Triangle-Free Graphs

Triply Existentially Complete Triangle-Free Graphs

A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+ 1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin [8, 9] asked whether k-existentially complete triangle-free graphs exist for every k. Here we present known and new constructions of 3existentially complete triangle-free ...

Measuring triangle-free graphs

Median Graphs and Triangle-Free Graphs

Let M(m,n) be the complexity of checking whether a graph G with m edges and n vertices is a median graph. We show that the complexity of checking whether G is triangle-free is at most M(m,m). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most O(m log n) + T (m log n, n), where T (m,n) is the complexity of finding all triangles of the graph. W...

Line graphs of triangle-free graphs

We present some results on the structure of line graphs of triangle-free graphs and discuss connections with homogeneity.

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph K⌊n/2⌋,⌈n/2⌉ contains more cycles than any other n-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and u...