Frustrated triangles
نویسندگان
چکیده
A triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges. We study the set Fn ⊂ [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, n3/2] cannot appear in Fn, and we characterise the graphs G with f (G) ∈ [0, n3/2]. More precisely, our main result is that, for each n ≥ 3, Fn contains two interlacing sequences 0 = a0 ≤ b0 ≤ a1 ≤ b1 ≤ · · · ≤ am ≤ bm ∼ n3/2 such that Fn ∩ (bt , at+1) = ∅ for all t , where the gaps are |bt −at+1| = (n−2)−t(t+1) and |at −bt | = t(t−1). Moreover, f (G) ∈ [at , bt ] if and 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 ifm ∈ [f (n), n 3 − f (n)], then m ∈ Fn where f (n) is asymptotically best possible with f (n) ∼ n3/2 for n even and f (n) ∼ √ 2n3/2 for n odd. Furthermore, we determine the graphs with the minimum number of frustrated triangles amongst those with n vertices and e ≤ n2/4 edges. © 2015 Elsevier B.V. All rights reserved.
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ورودعنوان ژورنال:
- Discrete Mathematics
دوره 338 شماره
صفحات -
تاریخ انتشار 2015