On the Implicit Graph Conjecture

The implicit graph conjecture states that every sufficiently small, hereditary graph class has a labeling scheme with a polynomial-time computable label decoder. We approach this conjecture by investigating classes of label decoders defined in terms of complexity classes such as P and EXP. For instance, GP denotes the class of graph classes that have a labeling scheme with a polynomial-time computable label decoder. Until now it was not even known whether GP is a strict subset of GR. We show that this is indeed the case and reveal a strict hierarchy akin to classical complexity. We also show that classes such as GP can be characterized in terms of graph parameters. This could mean that certain algorithmic problems are feasible on every graph class in GP. Lastly, we define a more restrictive class of label decoders using first-order logic that already contains many natural graph classes such as forests and interval graphs. We give an alternative characterization of this class in terms of directed acyclic graphs. By showing that some small, hereditary graph class cannot be expressed with such label decoders a weaker form of the implicit graph conjecture could be disproven.Comment: 13 pages, MFCS 201

Small But Unwieldy

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most $s \log n$. In other words, for every $s$, there is a small (even tiny) monotone class without universal graphs of size $n^s$. Prior to this result, it was not excluded that every small class has an almost linear universal graph, or equivalently a labeling scheme with labels of size $(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, ESA '07; Dujmovi\'{c} et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small monotone classes have unbounded twin-width, thus simultaneously disprove the already-refuted Small conjecture; but this time with a self-contained proof, not relying on elaborate group-theoretic constructions.Comment: 24 pages, 1 figure, shortened abstrac

Applications of entropy to extremal problems

The Sidorenko conjecture gives a lower bound on the number of homomorphisms from a bipartite graph to another graph. Szegedy [28] used entropy methods to prove the conjecture in some cases. We will refine these methods to also give lower bounds for the number of injective homomorphisms from a bipartite graph to another bipartite graph, and a lower bound for the number of homomorphisms from a k-partite hypergraph to another k-partite hypergraph, as well as a few other similar problems. Next is a generalisation of the Kruskal Katona Theorem [19, 17]. We are given integers k 4 we will make a lot of progress towards finding a solution. The next chapter is to do with Turán-type problems. Given a family of k-hypergraphs F, ex(n;F) is the maximum number of edges an F-free n-vertex k-hypergraph can have. We prove that for a rational r, there exists some finite family F of k-hypergraphs for which ex(n;F) = Ɵ(nk-r) if and only if 0 < r < k - 1 or r = k. The final chapter will deal with the implicit representation conjecture, in the special case of semi-algebraic graphs. Given a graph in such a family, we want to assign a name to each vertex in such a way that we can recover each edge based only on the names of the two incident vertices. We will first prove that one `obvious' way of storing the information doesn't work. Then we will come up with a way of storing the information that requires O(n1-E) bits per vertex, where E is some small constant depending only on the family