Generalized Girth Problems in Graphs and Hypergraphs

We study the asymptotic value of several extremal problems on graphs and hypergraphs, that arise as generalized notions of girth. Apart from being combinatorially natural questions, they are motivated by computational-theoretic applications. 1. An `-subgraph is a subgraph with ` edges per vertex, or equivalently, average degree 2`. What is the optimal upper bound S`(n, d), such that any graph of size n and density d must contain an `-subgraph of size at most S`(n, d)? The ` = 1 case coincides with the girth problem, and the answer is well known to be Θ(logd−1 n). For ` ≥ 2 we give nearly tight upper and lower bounds: ∀ > 0, Ω(n/d ` `−1 ) ≤ S`(n, d) ≤ O(n/d ` `−1− 1 `3−2`+1 − ) For example for ` = 2, every graph of size n and density d contains a subgraph of size O(n/d1.8− ) and average degree 4. We further improve the upper bound to O(n2− ), nearly meeting the lower bound, for graphs with bounded density or large girth, and conjecture it should hold for general graphs as well. 2. The `-girth of a graph is the size of its smallest subgraph with minimum degree `. What is the optimal upper bound g`(n, d), such that any graph of size n and density d has `-girth at most g`(n, d)? Erdős et al. [EFRS90] proved that g`(n, d) = O(n/d). We prove, ∀ > 0, Ω(n/d 2 `−2 ) ≤ g`(n, d) ≤ O(n/d 1+ 1 `−2− 1 `3−3`2+`+2 − ) For example for 3-girth, we get Ω(n/d3) ≤ g3(n, d) ≤ O(n/d1.8− ). 3. The S`(n, d) question is naturally posed on hypergraphs as well. We give upper bounds on the size of (2/3)-subgraphs in 3-uniform hypergraphs, progressing towards a conjecture of Feige [Fei08] that was raised in the context of efficient random 3CNF refutation.