The toughness of a non-complete graph G=(V,E) is defined as t(G)=min|S|c(G−S):S⊆V(G)andc(G−S)≥2,where c(G−S) is the number of components of G−S. By convention, t(Kn)=+∞. Haemers (2020) conjectured that every connected graph G of order n≥2 with the minimum degree δ satisfies t(G)≥μ2μn−δ,where μn and μ2 are the largest and the second smallest Laplacian eigenvalues of G, respectively. We give an independent proof of this conjecture and discuss its sharpness and several consequences.
For a sequence a1,a2,… in a semigroup, let FS(a1,a2,…) be the set of all sums ai1+⋯+aim, where m≥1 and i1<⋯<im. An infinite parallelepiped (IP) set is a set that contains FS(a1,a2,…) for some sequence a1,a2,…. Hindman’s Finite Sums Theorem asserts that the IP sets of natural numbers are partition regular: for each finite coloring of an IP set of natural numbers, there is a monochromatic IP subset. Furstenberg generalized Hindman’s theorem to arbitrary semigroups, but in this generality, the guaranteed monochromatic IP set may be a singleton.A proper infinite parallelepiped (IP) set in a semigroup is a set containing an infinite set of the form FS(a1,a2,…); equivalently, it is a set containing FS(a1,a2,…) for some injective sequence a1,a2,…. We provide a complete classification of the semigroups for which the proper IP sets are partition regular, and show that this property is equivalent to other fundamental notions of additive Ramsey theory.
For a graph F, let h_F(n,q) be the minimum number of copies of F in an n-vertex graph with ex(n,F)+q edges, where ex(n,F) is the maximum number of edges in an n-vertex F-free graph. Let c(n,F) be the minimum number of copies obtained by adding one edge to an extremal F-free graph. Mubayi's supersaturation conjecture predicts, under a stability hypothesis, that h_F(n,q)≥ q c(n,F). Ma and Yuan recently constructed stable graph counterexamples for every fixed q≥4; they asked whether the one-edge equality h_F(n,1)=c(n,F) might still hold for every graph F containing a cycle. We give a negative answer to their question. For each integer t≥6, let H_t be obtained from the t-vertex path by replacing each edge with a 3t-page book, using disjoint page vertices for different path edges. Then h_H_t(n,1)<c(n,H_t) for infinitely many values of n. Moreover, by taking t large, the ratio h_H_t(n,1)/c(n,H_t) can be made arbitrarily small along infinitely many values of n.
An r-uniform hypergraph is called t-union-free if any two distinct subsets of at most t edges have distinct union. The study of union-free hypergraphs has multiple origins and a long history, dating back to the works of Kautz and Singleton (1964) in coding theory, Bollobás and Erdős (1976) in combinatorics, and Hwang and Sós (1987) in group testing. Let U_t(n,r) denote the maximum number of edges in an n-vertex t-union-free r-uniform hypergraph. In this paper, we determine the asymptotic behavior of U_t(n,r), up to a lower order term, for almost all t≥ 3 and r≥ 3. This significantly advances the understanding of this extremal function, as previously, only the asymptotics of U_2(n,3) and U_2(n,4) were known. As a key ingredient of our proof, we establish the existence of near-optimal locally sparse induced hypergraph packings, which is of independent interest.
