We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.

We calculate some extremal and non-extremal four-point functions on the sphere of certain chiral primary operators for strings on AdS3 ×S3 × T . The computation is done for small values of the spacetime cross-ratio where global SL(2) and SU(2) descendants may be neglected in the intermediate channel. Ignoring also current algebra descendants, we find that in the non-extremal case the integrated...

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G)) ≤ n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G)) = n(G) + 1. An algorithm is given firstly to judge w...

For a graph G, let τ(G) be the decycling number of G and c(G) be the number of vertex-disjoint cycles of G. It has been proved that c(G)≤ τ(G)≤ 2c(G) for an outerplanar graph G. An outerplanar graph G is called lower-extremal if τ(G)= c(G) and upper-extremal if τ(G)= 2c(G). In this paper, we provide a necessary and sufficient condition for an outerplanar graph being upper-extremal. On the other...

We give a classification of four-circulant singly even self-dual [60, 30, d] codes for d = 10 and 12. These codes are used to construct extremal singly even self-dual [60, 30, 12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60, 30, 12] codes, we also construct extremal singly even self-dual [58...

A non-uniform hypergraph H = (V,E) consists of a vertex set V and an edge set E ⊆ 2V ; the edges in E are not required to all have the same cardinality. The set of all cardinalities of edges in H is denoted by R(H), the set of edge types. For a fixed hypergraph H, the Turán density π(H) is defined to be limn→∞maxGn hn(Gn), where the maximum is taken over all H-free hypergraphs Gn on n vertices ...

With the untimely passing of this still-young mathematician, the world has lost an “original”. Hunter Snevily came to the University of Illinois in the mid-1980s and became only my fifth Ph.D. student, completing his thesis in 1991. I was also young then and learned as much from him as he did from me. Hunter’s main work was in extremal set theory, where he made a significant contribution. As a ...

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A graph G = (V, E) is matching immune if there is no matching cut in G. We show that for any matching immune graph G, |E| ≥ ⌈3(|V | − 1)/2⌉. This bound is tight, as we define operations that construct, from a given vertex, exactly the class of matching immune graphs that attain the bound.

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by ...