Frustrated triangles

Abstract

a b s t r a c tA triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges.We study the set F n ⊂ [0,  n 3  ] of possible number of frustrated triangles f (G) in a graph G on n vertices. We prove that about two thirds of the numbers in [0, n 3/2 ] cannot appear in F n , and we characterise the graphs G with f (G) ∈ [0, n 3/2]. More precisely, our main result is that, for each n ≥ 3, F n contains two interlacing sequences 0and only if G can be obtained from a complete bipartite graph by flipping exactly t edges/nonedges. On the other hand, we show, for all n sufficiently large, that if m, then m ∈ F n where f (n) is asymptotically best possible with f (n) ∼ n 3/2 for n even and f (n) ∼ √ 2n 3/2 for n odd. Furthermore, we determine the graphs with the minimum number of frustrated triangles amongst those with n vertices and e ≤ n 2 /4 edges.

Terminal-pairability in complete bipartite graphs with non-bipartite demands

“…The problem is also very closely related to the integer multicommodity flow problems and the theory of graph immersions, each with their own terminologies. In this paper from now on we use the terminology of terminal-pairability, as other papers [7,13,12,14,25,18,26,27,23,28,19] about sufficient conditions do.…”

The dimension of the negative cycle vectors of a signed graph

“…The vectors for K 3 are (0), (1) (from the balanced and unbalanced triangle). The vectors for K 4 are (0, 0), (2, 2), (4, 0)…”

“…Finally, which vectors in the convex hull are actually the vectors of signed graphs? Recently Kittipassorn and Mészáros [1] gave strong restrictions on the number of negative triangles in a signed K n . Again, this provides a step towards that answer.…”

“…The vectors for K 3 are (0),(1) (from the balanced and unbalanced triangle). The vectors for K 4 are (0, 0), (2, 2), (4, 0)(the all-positive graph, one negative edge, and two nonadjacent negative edges).…”

Negative Circles in Signed Graphs: A Problem Collection

“…Characterize the set of numbers of negative circles of some fixed length of all signatures of K n ; that is, {c − l (K n , σ) : σ is a signature of K n } for some fixed l, 3 ≤ l ≤ n. Ans. Very recently there are remarkably strong results on the possible numbers of negative triangles (Kittipassorn and Mészáros [7]). I am not aware of any results about longer circles.…”