One of the best-known results in spectral graph theory is the inequality of Hoffman \[ χ\left( G\right) \geq1-\frac{λ\left( G\right) }{λ_{\min }\left( G\right) }, \] where $χ\left( G\right) $ is the chromatic number of a graph $G$ and $λ\left( G\right) ,$ $λ_{\min}\left( G\right) $ are the largest and the smallest eigenvalues of its adjacency matrix.
In this note Hoffman's inequality is extended to weighted uniform $r$-graphs for every even $r$.
Using the Filmus-Golubev-Lifshitz method to bound the independence number of a hypergraph, we solve some problems concerning multiply intersecting families with biased measure. Among other results we obtain a stability result of a measure version of the Erdos-Ko-Rado theorem for multiply intersecting families.
One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Turán problem for (hyper-)graphs, can be encoded as special cases of thi…
The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from A. J. Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs less than $\sqrt{2+\sqrt{5}}$. The program consists of two aspects: finding all the possible limit points of $M$-spectral radii of graphs and detecting all the connected graphs whose $M$-spectral radius does not exceed a fixed limit point. In this paper, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless…
Hoffman proved that for a simple graph $G$, the chromatic number $χ(G)$ obeys $χ(G) \le 1 - \frac{λ_1}{λ_{n}}$ where $λ_1$ and $λ_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$ respectively. Lovász later showed that $χ(G) \le 1 - \frac{λ_1}{λ_{n}}$ for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lovász's theorem, then extend the technique to derive generalizations of Hoffman's theorem when allowed a certain proportion of edge-conflicts. Using this result, we show that if a 3-uniform hypergraph is 2-colorable, th…
Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With this note I hope to clarify the history of this bound and some of its generalizations.
We extend Hoffman's coclique bound for regular digraphs with the property that its adjacency matrix is normal, and discuss cocliques attaining the inequality. As a consequence, we characterize skew-Bush-type Hadamard matrices in terms of digraphs. We present some normal digraphs whose vertex set is decomposed into disjoint cocliques attaining the bound. The digraphs provided here are relation graphs of some nonsymmetric association schemes.
Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and…
Let $χ(G)$ denote the chromatic number of a graph and $χ_v(G)$ denote the vector chromatic number. For all graphs $χ_v(G) \le χ(G)$ and for some graphs $χ_v(G) \ll χ(G)$. Galtman proved that Hoffman's well-known lower bound for $χ(G)$ is in fact a lower bound for $χ_v(G)$. We prove that two more spectral lower bounds for $χ(G)$ are also lower bounds for $χ_v(G)$. We then use one of these bounds to derive a new characterization of $χ_v(G)$.
Hofmman's bound on the chromatic number of a graph states that $χ\geq 1 - \frac {λ_1} {λ_n}$. Here we show that the same bound, or slight modifications of it, hold for several graph parameters related to the chromatic number: the vector coloring number, the $ψ$-covering number and the $λ$-clustering number.
The $k$-th $p$-power of a graph $G$ is the graph on the vertex set $V(G)^k$, where two $k$-tuples are adjacent iff the number of their coordinates which are adjacent in $G$ is not congruent to 0 modulo $p$. The clique number of powers of $G$ is poly-logarithmic in the number of vertices, thus graphs with small independence numbers in their $p$-powers do not contain large homogenous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of Alon and Lubetzky up to a factor of 2. For precise bounds on some graphs, we ap…
A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $ν_u$ of the Perron eigenvector $\mathbfν$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity …
In this paper, we consider upper bounds on the size of transitive subtournaments in a digraph. In particular, we give an analogy of Hoffman's bound for the size of cocliques in a regular graph. Furthermore, we partially improve the Hoffman type bound for doubly regular tournaments by using the technique of Greaves and Soicher for strongly regular graphs [4], which gives a new application of block intersection polynomials.
The notion of duality is a key element in understanding the interplay between the stability and chromatic numbers of a graph. This notion is a central aspect in the celebrated theory of perfect graphs, and is further and deeply developed in the context of the Lovász theta function and its equivalent characterizations and variants. The main achievement of this paper is the introduction of a new family of norms, providing upper bounds for the stability number, that are obtained from duality from the norms motivated by Hoffman's lower bound for the chromatic number and which achieve the (compleme…
Generalizing work on graphs, Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs, attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\geq 7$. Furthermore, we show that the squid hyp…
Hypergraphs are an invaluable tool to understand many hidden patterns in large data sets. Among many ways to represent hypergraph, one useful representation is that of weighted clique expansion. In this paper, we consider this representation for non-uniform hypergraphs. We generalize the spectral results for uniform hypergraphs to non-uniform hypergraphs and show that they extend in a natural way. We provide a bound on the largest eigenvalue with respect to the average degree of neighbours of a vertex in a graph. We also prove an inequality on the boundary of a vertex set in terms of the large…
We study the size of the shadow of $k$-uniform hypergraphs with bounded degree. Lower bounds on the ratio of the size of the shadow and the size of the hypergraph are given as a function of the degree bound and $k$. We show that cliques are extremal for a long range of degree bounds, but not for every bound. We give a general, but not sharp lower bound on the shadow ratio and show, that sometimes we can get extremal hypergraphs by deleting disjoint maximal matchings from a clique.
The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
We give sharp upper bounds for the ordinary spectral radius and signless Laplacian spectral radius of a uniform hypergraph in terms of the average $2$-degrees or degrees of vertices, respectively, and we also give a lower bound for the ordinary spectral radius. We also compare these bounds with known ones.
Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the…
We give an upper bound on the list chromatic number of a 2-colorable hypergraph which generalizes the bound of Schauz on $k$-partite $k$-uniform hypergraphs. It makes sense for sparse hypergraphs: in particular we show that a $k$-uniform $k$-regular hypergraph has the list chromatic number 2 for $k \geq 4$. Also we obtain both lower and upper bound on the list chromatic number of a complete $s$-uniform 2-colorable hypergraph in the vein of Erd{\H o}s--Rubin--Taylor theorem.
Let $\mathcal{H}$ be a uniform hypergraph. Let $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ be the adjacency tensor and the signless Laplacian tensor of $\mathcal{H}$, respectively. In this note we prove several bounds for the spectral radius of $\mathcal{A(H)}$ and $\mathcal{Q(H)}$ in terms of the degrees of vertices of $\mathcal{H}.$
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate tha…
Lower bounds for the first and the second eigenvalue of uniform hypergraphs which are regular and linear are obtained. One of these bounds is a generalization of the Alon-Boppana Theorem to hypergraphs.
More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful t…
A hypergraph is said to be properly 2-colorable if there exists a 2-coloring of its vertices such that no hyperedge is monochromatic. On the other hand, a hypergraph is called non-2-colorable if there exists at least one monochromatic hyperedge in each of the possible 2-colorings of its vertex set. Let m(n) denote the minimum number of hyperedges in a non-2-colorable n-uniform hypergraph. Establishing the lower and upper bounds on m(n) is a well-studied research direction over several decades. In this paper, we present new constructions for non-2-colorable n-uniform hypergraphs. These construc…
The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. In a recent paper we have shown that these bounds are unavoidable for $3$-uniform hypergraphs. In this paper we exten…
A fundamental problem in pattern avoidance is describing the asymptotic behavior of the extremal function and its generalizations. We prove an equivalence between the asymptotics of the graph extremal function for a class of bipartite graphs and the asymptotics of the matrix extremal function. We use the equivalence to prove several new bounds on the extremal functions of graphs. We develop a new method to bound the extremal function of hypergraphs in terms of the extremal function of their associated multidimensional matrices, improving the bound of the extremal function of $d$-permutation hy…
Let $G$ be a connected uniform hypergraphs with maximum degree $Δ$, spectral radius $λ$ and minimum H-eigenvalue $μ$. In this paper, we give some lower bounds for $Δ-λ$, which extend the result of [S.M. Cioabă, D.A. Gregory, V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory, Ser. B 97 (2007) 483-486] to hypergraphs. Applying these bounds, we also obtain a lower bound for $Δ+μ$.
We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph.
We study the maximum number of hyperedges in a 3-uniform hypergraph on $n$ vertices that does not contain a Berge cycle of a given length $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order $O(k^2n^{1+1/k})$, improving the upper bound of Györi and Lemons by a factor of $Θ(k^2)$. Similar bounds are shown for linear hypergraphs.
For $0\leq α< 1$, the $\mathcal{A}_α$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_α(G):=α\mathcal{D}(G)+(1-α) \mathcal{A}(G)$, where $\mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $\mathcal{A}_α$-spectral radius and an average degree $\frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the …
The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao. Pri…
The famous Dirac's Theorem gives an exact bound on the minimum degree of an $n$-vertex graph guaranteeing the existence of a hamiltonian cycle. We prove exact bounds of similar type for hamiltonian Berge cycles in $r$-uniform, $n$-vertex hypergraphs for all $3\leq r< n$. The bounds are different for $r<n/2$ and $r\geq n/2$. We also give bounds on the minimum degree guaranteeing existence of Berge cycles of length at least $k$ in such hypergraphs; the bounds are exact for all $k\geq n/2$.
In this paper, we consider maximum possible value for the sum of cardinalities of hyperedges of a hypergraph without a Berge $4$-cycle. We significantly improve the previous upper bound provided by Gerbner and Palmer. Furthermore, we provide a construction that slightly improves the previous lower bound.
We obtain new lower bounds for the independence number of $K_r$-free graphs and linear $k$-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we also give a new short proof of the main result of \cite{CT91} using this approach. As byproducts, we also obtain some non-trivial identities involving binomial coefficients.
We consider the problem of bounding the number of permutations $σ\in S_n$ that avoid a fixed permutation $π\in S_k$ in specific indices given by a $k$-uniform hypergraph $Λ$. We obtain relatively sharp bounds in the case where $Λ$ is a random hypergraph, and find bounds in the case where $Λ$ contains many large cliques. Along the way, we prove a supersaturation version of Füredi-Hajnal, which may be of independent interest.
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
For $r\geq 3$, let $f_r\colon [0,\infty)\to [1,\infty)$ be the unique analytic function such that $f_r({k\choose r})={k-1\choose r-1}$ for any $k\geq r-1$. We prove that the spectral radius of an $r$-uniform hypergraph $H$ with $e$ edges is at most $f_r(e)$. The equality holds if and only if $e={k\choose r}$ for some positive integer $k$ and $H$ is the union of a complete $r$-uniform hypergraph $K_k^r$ and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs.
In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains $Ω(\frac{2^d}{\sqrt{d}})$ $n\choose 2d$ crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lo…
Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).
The Lovász Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general theorem for colouring hypergraphs that in many instances matches or slightly improves upon the bounds obtained using the Lovász Local Lemma. Moreover, the theorem directly shows that there are exponentially many colourings. The elementary and self-contained proof is inspired by a recent result for nonrepetitive colourings by Rosenfeld [2020]. We apply our general t…
We prove that every 52-connected line graph of a rank 3 hypergraph is Hamiltonian. This is the first result of this type for hypergraphs of bounded rank other than ordinary graphs.
