Extremal problems for multigraphs

• An (n,s,q)-graph is an n-vertex multigraph in which every s-set of vertices spans at most q edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in (n,s,q)-graphs. We give a general lower bound construction for this problem for many pairs (s,q), which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the (s,q)=(4,6a+3) case of the problem (for a≥2); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for ‘sparse’ multigraphs (i.e. when q≤2(s2)). Finally we introduce some tools that are likely to be useful for attacking the problem in general.

Ämnesord

NATURVETENSKAP  -- Matematik -- Diskret matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Discrete Mathematics (hsv//eng)

Nyckelord

Multigraphs

Transcendental numbers

Turán-type problems

Mathematics

matematik

Publikations- och innehållstyp

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