Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $ν(H)$, the vertex cover number is at most $(r-1)ν(H)$. This far reaching generalization of König's theorem is only known to be true for $r\leq 3$, or $ν(G)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $α(G)$, there exists at most $(r-1)α(G)$ monochromatic connected subgraphs which cover the vertex set of $G$.
We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.
A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points satisfying (i) every two blocks intersect each other in $λ$ points for a fixed $λ< v$ (ii) there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if all the blocks of $\mathcal{D}$ have the same size (or equivalently, every point has the same replication number) and every two blocks intersect each other in $λ$ points. The only known construction of a Ryser design is via block complementation of a symmetric design also known as the Rys…
Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integer…
A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $λ$ points (and $λ< v$ is a fixed number) and there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if every point of $\mathcal{D}$ has the same replication number (or equivalently, all the blocks have the same size) and every two blocks intersect each other in $λ$ points. The only known construction of a Ryser design is via block complementation of a symmetric design. Such a Ryser d…
In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conje…
Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph the vertex cover number is at most $r-1$ times the matching number. This conjecture is only known to be true for $r\leq 3$. For intersecting hypergraphs, Ryser's Conjecture reduces to saying that the edges of every $r$-partite intersecting hypergraph can be covered by $r-1$ vertices. This special case of the conjecture has only been proven for $r \leq 5$. It is interesting to study hypergraphs which are extremal in Ryser's Conjecture i.e, those hypergraphs for which the vertex cover number is exactly $r-1$ times the matc…
A famous conjecture (usually called Ryser's conjecture) that appeared in the Ph.D thesis of his student, J.~R.~Henderson [15], states that for an $r$-uniform $r$-partite hypergraph $\mathcal{H}$, the inequality $τ(\mathcal{H})\le(r-1)\cdot ν(\mathcal{H})$ always holds. This conjecture is widely open, except in the case of $r=2$, when it is equivalent to K\H onig's theorem [18], and in the case of $r=3$, which was proved by Aharoni in 2001 [3]. Here we study some special cases of Ryser's conjecture. First of all the most studied special case is when $\mathcal{H}$ is intersecting. Even for this …
Ryser conjectured that $τ\le(r-1)ν$ for $r$-partite hypergraphs, where $τ$ is the covering number and $ν$ is the matching number. We prove this conjecture for $r\le9$ in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser's conjecture which specified that each $r$-partite hypergraph should have a cover of size $(r-1)ν$ of a particular form. We provide a counterexample to Aharoni's conjecture with $r=13$ and $ν=1$. We also report a number of computational results. For $r=7$, we …
A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been studied extensively. It was recently proved by Haxell, Narins and Szabó that all $3$-Ryser hypergraphs with matching number $ν> 1$ are essentially obtained by taking $ν$ disjoint copies of intersecting $3$-Ryser hypergraphs. Abu-Khazneh showed that such a characterisation is false for $r = 4$ by giving a computer generated example of a $4$-Ryser hypergraph w…
A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}…
We show that an $n\times n$ circulant Hadamard matrix must satisfy a family of congruence equations that have solutions only when $n \leq 4$, proving Ryser's 1963 conjecture that no such matrices exist for $n>4$.
A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $λ$ points (and $λ< v$ is a fixed number) and there are at least two block sizes. A design $\mathcal{D}$ is called a symmetric design, if every point of $\mathcal{D}$ has the same replication number (or equivalently, all the blocks have the same size) and every two blocks intersect each other in $λ$ points. The only known construction of a Ryser design is via block complementation of a symmetric design. Such a Ryser d…
Let $τ(\mathcal{H})$ be the cover number and $ν(\mathcal{H})$ be the matching number of a hypergraph $\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\mathcal{H}$ satisfies the inequality $τ(\mathcal{H}) \leq (r-1) ν(\mathcal{H})$. This conjecture is open for all $r \ge 4$. For intersecting hypergraphs, namely those with $ν(\mathcal{H})=1$, Ryser's conjecture reduces to $τ(\mathcal{H}) \leq r-1$. Even this conjecture is extremely difficult and is open for all $ r \ge 6$. For infinitely many $r$ there are examples of intersecting $r$-partite hypergraphs with $τ(\mathcal{H})=r…
Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply König's Theorem and every bipartite graph is a witness for its tightness. The conjecture has also been proven for $r = 3$ by Aharoni using topological methods, but the proof does not give information on the extremal $3$-uniform hypergraphs. Our goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem. Our proof of this characterization is also based on topological machi…
Two prominent conjectures by Herbert J. Ryser have been falsely attributed to a somewhat obscure conference proceedings that he wrote in German. Here we provide a translation of that paper and try to correct the historical record at least as far as what was conjectured in it. The two conjectures relate to transversals in Latin squares of odd order and to the relationship between the covering number and the matching number of multipartite hypergraphs.
A famous conjecture of Ryser states that any $r$-partite set system has transversal number at most $r-1$ times their matching number. This conjecture is only known to be true for $r\leq3$ in general, for $r\leq5$ if the set system is intersecting, and for $r\leq9$ if the intersecting set system is linear. In this note, we deal with Ryser's Conjecture for intersecting $r$-partite linear systems; that is, if $τ$ is the transversal number for an intersecting $r$-partite linear system, then Ryser's Conjecture states that $τ\leq r-1$. If this conjecture is true, this is known to be sharp for $r$ fo…
A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i. The aim of the paper is twofold - to put order in the multitude of conjectures that relate to this concept (some of them first presented here), and to present some partial results on one of these conjectures, that seems central among them.
A well-known conjecture, often attributed to Ryser, states that the cover number of an $r$-partite $r$-uniform hypergraph is at most $r - 1$ times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is $t$-intersecting, conjecturing that the cover number $τ(\mathcal{H})$ of such a hypergraph $\mathcal{H}$ is at mos…
A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order $n\times n$ contains a transversal of order $n-1$. In this paper we prove the existence of a transversal of order $n-O(\log{n}/\log{\log{n}})$, i…
In this note we prove the conjecture of \cite{HaWiWi} that every bipartite multigraph with integer edge delays admits an edge colouring with $d+1$ colours in the special case where $d=3$. A connection to the Brualdi-Ryser-Stein conjecture is discussed.
In this paper, we present a minimal counterexample to a conjecture of Perles that answers a question of Haase and Ziegler. The example is a simple 4-polytope that has an induced 3-connected 3-regular subgraph, whose graph complement is connected. This subgraph is planar and not the graph of a facet of the polytope.
Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-mi…
We address an old (1977) conjecture of a subset of the authors (a variant of Ryser's conjecture): in every r-coloring of the edges of a biclique [A,B] (complete bipartite graph), the vertex set can be covered by the vertices of at most 2r-2 monochromatic connected components. We reduce this conjecture to design-like conjectures, where the monochromatic components of the color classes are bicliques [X,Y] with nonempty blocks X and Y. We prove this conjecture for r<6. We show that the width (the number of bicliques) in every color class of any spanning r-coloring is at most 2^{r-1} (and this is …
We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r-4 for all but finitely many r. This answers a question of Abu-Khazneh, Barat, Pokrovskiy and Szabo, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r.
Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger conjectured that $f(n)=n+1$ for every $n>1$. Clemens and Ehrenm{ü}ller proved that $f(n) \le \frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely $f(n) \le \lceil \frac{3}{2}n \rceil+1$.
We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.
For a 2-connected graph $G$ on $n$ vertices and two vertices $x,y\in V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if there are at least $\frac{n-1}{2}$ vertices in $V(G)\backslash \{x,y\}$ of degree at least $k$. This strengthens a well-known theorem due to Erdős and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with $n$ vertices contains a cycle of length at least $2k$ if it has at least $\frac{n}{2}+k$ vertices of degree at least $k$. This confirms a 1975 conjecture made by Woodall. As another applications, we obtain some re…
Reed's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisfies $χ\leq \lceil\frac 12(Δ+1+ω)\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster than the best known algorithm…
Every properly colored graph with $χ(G)=k$ colors has edge-disjoint Kempe "backbones", Kempe chains anchored by color-critical vertices for each pair of colors. Certain color permutations arrange these backbones into a clique-like structure, a strengthening of strong immersions of complete graphs. This strengthened immersion is suggested as a template for identifying the disjoint subgraphs comprising Hadwiger's conjectured $K_k$ minor present in $k$-chromatic graphs.
In a latin square of order $n$, a near transversal is a collection of $n-1$ cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.
Resolving a conjecture of Bollobás and Erdős, Gyárfás proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by showing that such $G$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ consecutive odd lengths. Along the way, combining extremal and structural tools, we prove a stronger statement that every graph of chromatic number $k+1\geq 7$ contains $k$ cycles of consecutive lengths, except that some block is $K_{k+1}$. As corollaries, this confirms a conjecture of Verstraëte and answers…
A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gyárfás: if the edges of a complete graph K are colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Király proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \…
Given an integer $r$ and a vector $\vec{a}=(a_1, \ldots ,a_p)$ of positive numbers with $\sum_{i \le p} a_i=r$, an $r$-uniform hypergraph $H$ is said to be $\vec{a}$-partitioned if $V(H)=\bigcup_{i \le p}V_i$, where the sets $V_i$ are disjoint, and $|e \cap V_i|=a_i$ for all $e \in H,~~i \le p$. A $\vec{1}$-partitioned hypergraph is said to be $r$-partite. Let $t(\vec{a})$ be the maximum, over all intersecting $\vec{a}$-partitioned hypergraphs $H$, of the minimal size of a cover of $H$. A famous conjecture of Ryser is that $t(\vec{1})\le r-1$. Tuza conjectured that if $r>2$ then $t(\vec{a})=r$…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos (Combinatorica, 2006), Simonyi, Tardif, and Zsbán (The Electronic Journal of Combinatorics, 2013), and Chen (Journal of Combinatorial Theory, Series A, 2011). As a consequence of the generalization of Chen's theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisf…
We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-unif…
Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher. One extension of the conjecture that has been used by several researchers is to consider vertex-weighted digraphs. In this paper we introduce a version of the conjecture for arc-weighted digraphs. We prove the conjecture in the special case of arc-weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be s…
A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face $α$ the maximal color appears exactly once on the vertices of $α$. Fabrici and Göring proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color theorem. Wendland later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. Thus we conclude that facial unique-maximum chromatic number of the sphere is five.
Reed's well-known $ω$, $Δ$, $χ$ conjecture proposes that every graph satisfies $χ\leq \lceil \frac 12(Δ+1+ω)\rceil$. The second author formulated a {\em local strengthening} of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent vertices. We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional r…
Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture holds for the alternating groups when the prime p is odd.
We propose a combinatorial approach to the following strengthening of Gal's conjecture: $γ(Δ)\ge γ(E)$ coefficientwise, where $Δ$ is a flag homology sphere and $E\subseteq Δ$ an induced homology sphere of codimension $1$. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag $d$-polytopes: $h_1(Δ) h_i(Δ) \ge (d-i+1)h_{i-1}(Δ) + (i+1) h_{i+1}(Δ)$ for all $0\le i\le d$.
