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- W3083861195 abstract "An $(n,s,q)$-graph is an $n$-vertex multigraph where every set of $s$ vertices spans at most $q$ edges. In this paper, we determine the maximum product of the edge multiplicities in $(n,s,q)$-graphs if the congruence class of $q$ modulo ${schoose 2}$ is in a certain interval of length about $3s/2$. The smallest case that falls outside this range is $(s,q)=(4,15)$, and here the answer is $a^{n^2+o(n^2)}$, where $a$ is transcendental assuming Schanuel's conjecture. This could indicate the difficulty of solving the problem in full generality. Many of our results can be seen as extending work by Bondy and Tuza [J. Graph Theory, 25 (1997), pp. 267--275] and Füredi and Kündgen [J. Graph Theory, 40 (2002), pp. 195--225] about sums of edge multiplicities to the product setting. We also prove a variety of other extremal results for $(n,s,q)$-graphs, including product-stability theorems. These results are of additional interest because they can be used to enumerate $(n,s,q)$-graphs. Our work therefore extends many classical enumerative results in extremal graph theory beginning with the Erdös--Kleitman--Rothschild theorem [Asymptotic enumeration of $Ksb{n}$-free graphs, in Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, Atti dei Convegni Lincei 17, Accad. Naz. Lincei, Rome, 1976, pp. 19--27] to multigraphs." @default.
- W3083861195 created "2020-09-14" @default.
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- W3083861195 date "2020-01-01" @default.
- W3083861195 modified "2023-10-17" @default.
- W3083861195 title "Extremal Theory of Locally Sparse Multigraphs" @default.
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- W3083861195 doi "https://doi.org/10.1137/19m1237612" @default.
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