The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set {1,2,3,4,5}. We show that for regular graphs it is sufficient to use weights 1, 2, 3, 4. Moreover, we prove the conjecture to hold for every

Existence of a perfect matching in a random bipartite digraph with bipartition (V1,V2), |Vi|=n, is studied. The graph is generated in two rounds of random selections of a potential matching partner such that the average number of selections made by each vertex overall is below 2. More precisely, in the first round each vertex chooses a potential mate uniformly at random, and independently of all vertices

Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant δ>0 such that for all c sufficiently large, there exist graphs G and H with chromatic number at least (1+δ)c for which χ(G×H)≤c.

A well known generalisation of Dirac's theorem states that if a graph G on n≥4k vertices has minimum degree at least n/2 then G contains a 2-factor consisting of exactly k cycles. This is easily seen to be tight in terms of the bound on the minimum degree. However, if one assumes in addition that G is Hamiltonian it has been conjectured that the bound on the minimum degree may be relaxed. This was

The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs. A theorem of Rödl asserts that if an n-vertex graph is non-universal then it contains an

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph K2n, there exists a decomposition of K2n into isomorphic rainbow spanning trees. This settles conjectures of Brualdi–Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.

This paper proves that every planar graph G contains a matching M such that the Alon-Tarsi number of G−M is at most 4. As a consequence, G−M is 4-paintable, and hence G itself is 1-defective 4-paintable. This improves a result of Cushing and Kierstead (2010) [5], who proved that every planar graph is 1-defective 4-choosable.

For k≥1 and n≥2k, the Kneser graph KG(n,k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of KG(n,k) is n−2k+2. Schrijver constructed a vertex-critical subgraph SG(n,k) of KG(n,k) with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges

Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one

We prove that every 3-edge-connected graph G has a 3-flow ϕ with the property that |supp(ϕ)|≥56|E(G)|. The graph K4 demonstrates that this 56 ratio is best possible; there is an infinite family where 56 is tight.

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6], the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2], we proved that every cycle has this property. Here we give a common

Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they

It is proved that every connected graph G on n vertices with χ(G)≥4 has at most k(k−1)n−3(k−2)(k−3) k-colourings for every k≥4. Equality holds for some (and then for every) k if and only if the graph is formed from K4 by repeatedly adding leaves. This confirms (a strengthening of) the 4-chromatic case of a long-standing conjecture of Tomescu [29]. Proof methods may be of independent interest. In particular

Let S be an n-element set and F⊂(Sk) an intersecting family. Improving earlier results it is proved that for n>72k there is an element of S that is contained in all but (n−3k−2) members of F. One of the main ingredients of the proof is the following statement. If G⊂(Sk) is intersecting, |G|≥(n−2k−2) and n≥72k then there is an element of S that is contained in more than half of the members of G.

For integers k≥3 and 1≤ℓ≤k−1, we prove that for any α>0, there exist ϵ>0 and C>0 such that for sufficiently large n∈(k−ℓ)N, the union of a k-uniform hypergraph with minimum vertex degree αnk−1 and a binomial random k-uniform hypergraph G(k)(n,p) with p≥n−(k−ℓ)−ϵ for ℓ≥2 and p≥Cn−(k−1) for ℓ=1 on the same vertex set contains a Hamiltonian ℓ-cycle with high probability. Our result is best possible up

A recent conjecture of the author and Teng Fang states that there are only finitely many finite simple groups with no cubic graphical regular representation. In this paper, we make crucial progress towards this conjecture by giving an affirmative answer for groups of Lie type of large rank.

We prove that every strongly 1050t-connected tournament contains all possible 1-factors with at most t components and this is best possible up to constant. In addition, we can ensure that each cycle in the 1-factor contains a prescribed vertex. This answers a question by Kühn, Osthus, and Townsend. Indeed, we prove more results on partitioning tournaments. We prove that a strongly Ω(k4tq)-connected

Given two k-graphs H and F, a perfect F-packing in H is a collection of vertex-disjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F-packing is simply a perfect matching. For a given fixed F, it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F-packing is NP-complete. Indeed, if k≥3, the corresponding

We disprove a conjecture of Frank [3] stating that each weakly 2k-connected graph has a k-vertex-connected orientation. For k≥3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

A graph G is k-ordered if for any distinct vertices v1,v2,…,vk∈V(G), it has a cycle through v1,v2,…,vk in order. Let f(k) denote the minimum integer so that every f(k)-connected graph is k-ordered. The first non-trivial case of determining f(k) is when k=4, where the previously best known bounds are 7≤f(4)≤40. We prove that in fact f(4)=7.

Twenty years ago Bondy and Vince conjectured that for any nonnegative integer k, except finitely many counterexamples, every graph with k vertices of degree less than three contains two cycles whose lengths differ by one or two. The case k≤2 was proved by Bondy and Vince, which resolved an earlier conjecture of Erdős et al. In this paper we confirm this conjecture for all k.

Assume k is a positive integer, λ={k1,k2,…,kq} is a partition of k and G is a graph. A λ-assignment of G is a k-assignment L of G such that the colour set ⋃v∈V(G)L(v) can be partitioned into q subsets C1∪C2…∪Cq and for each vertex v of G, |L(v)∩Ci|=ki. We say G is λ-choosable if for each λ-assignment L of G, G is L-colourable. It follows from the definition that if λ={k}, then λ-choosable is the same

A key theorem in algorithmic graph-minor theory is a min-max relation between the treewidth of a graph (i.e., the minimum width of a tree-decomposition) and the maximum size of a grid minor. This min-max relation is a keystone of the graph minor theory of Robertson and Seymour, which ultimately proves Wagner's Conjecture about properties of minor-closed graphs. Demaine and Hajiaghayi proved a remarkable

In 1998, Reed conjectured that every graph G satisfies χ(G)≤⌈12(Δ(G)+1+ω(G))⌉, where χ(G) is the chromatic number of G, Δ(G) is the maximum degree of G, and ω(G) is the clique number of G. As evidence for his conjecture, he proved an “epsilon version” of it, i.e. that there exists some ε>0 such that χ(G)≤(1−ε)(Δ(G)+1)+εω(G). It is natural to ask if Reed's conjecture or an epsilon version of it is true

In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph Kn. In this paper, we investigate the minimum genus embeddings of the complete 3-uniform hypergraphs Kn3. Embeddings of a hypergraph H are defined as the embeddings of its associated Levi graph LH with vertex set V(H)⊔E(H), in which v∈V(H) and e∈E(H) are adjacent if and

We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that “sufficiently large” graphs of fixed diameter and degree must be “good” expanders. We prove this statement for various definitions of “sufficiently

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem

Let M be a 3-connected matroid, and let N be a 3-connected minor of M. We say that a pair {x1,x2}⊆E(M) is N-detachable if one of the matroids M/x1/x2 or M\x1\x2 is both 3-connected and has an N-minor. This is the first in a series of three papers where we describe the structures that arise when M has no N-detachable pairs. In this paper, we prove that if no N-detachable pair can be found in M, then

Every 9-regular graph (possibly with multiple edges) with odd edge-connectivity >5 can be edge-decomposed into three 3-factors. If Tutte's 3-flow conjecture is true, it also holds for all 9-regular graphs with odd edge-connectivity 5, but not with odd edge-connectivity 3. It holds for all planar 2-edge-connected 9-regular graphs, an equivalent version of the 4-color theorem for planar graphs. We address

An edge coloring of the n-vertex complete graph, Kn, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for n large and every k with k≤2n/4300, the number of Gallai colorings of Kn that use at most k given colors is ((k2)+on(1))2(n2). Our result is asymptotically best possible and implies that, for

The number of cycles in a graph containing any fixed edge and also containing all vertices of odd degree is odd if and only if all vertices have even degree. If all vertices have even degree this is a theorem of Shunichi Toida. If all vertices have odd degree it is Andrew Thomason's extension of Smith's theorem.

A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K5 or K3,3. Wagner proved in 1937 that if a graph other than K5 does not contain any subdivision of K3,3 then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of

The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K5. This conjecture was proved by Ma and Yu for graphs containing K4−, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations

We use K4− to denote the graph obtained from K4 by removing an edge, and use TK5 to denote a subdivision of K5. Let G be a 5-connected nonplanar graph and {x1,x2,y1,y2}⊆V(G) such that G[{x1,x2,y1,y2}]≅K4− with y1y2∉E(G). Let w1,w2,w3∈N(y2)−{x1,x2} be distinct. We show that G contains a TK5 in which y2 is not a branch vertex, or G−y2 contains K4−, or G has a special 5-separation, or G−{y2v:v∉{w1,w2

Let G be a 5-connected nonplanar graph and let x1,x2,y1,y2∈V(G) be distinct, such that G[{x1,x2,y1,y2}]≅K4− and y1y2∉E(G). We show that one of the following holds: G−x1 contains K4−, or G contains a K4− in which x1 is of degree 2, or G contains a TK5 in which x1 is not a branch vertex, or {x2,y1,y2} may be chosen so that for any distinct z0,z1∈N(x1)−{x2,y1,y2}, G−{x1v:v∉{z0,z1,x2,y1,y2}} contains TK5

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field F1 and a field extension F2, if F1-representability is not equivalent to F2-representability, then each F1-representable matroid is a minor of a F2-representable excluded minor for F1-representability.

We consider the Turán-type problem of bounding the size of a set M⊆F2n that does not contain a linear copy of a given fixed set N⊆F2k, where n is large compared to k. An Erdős-Stone type theorem [5] in this setting gives a bound that is tight up to a o(2n) error term; our first main result gives a stability version of this theorem, showing that such an M that is close in size to the upper bound in

We introduce a class of plane graphs called weak near-triangulations, and prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on n>6 vertices has a dominating set of size at most 17n/53. This improves the bound n/3 obtained by Matheson and Tarjan.

We show that for every η>0 there exists n0 such that for every odd n≥n0 each 3-colouring of edges of a graph G with (4+η)n and minimum degree larger than (7/2+2η)n leads to a monochromatic cycle of length n. This result is, up to η terms, best possible.

Let G(n,r,s) denote a uniformly random r-regular s-uniform hypergraph on n vertices, where s is a fixed constant and r=r(n) may grow with n. An ℓ-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely ℓ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When r,s≥3 are fixed integers, we establish a threshold result for the property of

We give a complete answer to the GFR conjecture, proposed by Conder, Doyle, Tucker and Watkins: “All but finitely many Frobenius groups F=N⋊H with a given complement H have a GFR, with the exception when |H| is odd and N is Abelian but not an elementary 2-group”. Actually, we prove something stronger, we enumerate asymptotically GFRs; we show that, besides the exceptions listed above, as |N| tends

The inverse eigenvalue problem of a given graph G is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph. Electron

The τ=2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special

A tournament T=(V,A) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In the first paper we have given a

We prove that Gn,p=c/n w.h.p. has a k-regular subgraph if c is at least e−Θ(k) above the threshold for the appearance of a subgraph with minimum degree at least k; i.e. a non-empty k-core. In particular, this pins down the threshold for the appearance of a k-regular subgraph to a window of size e−Θ(k).

We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let F be a finite field. The first conjecture states that: the branch-width of any F-representable N-fragile matroid is bounded by a function depending only upon F and N. The second conjecture states that: if a matroid M2 is obtained from a matroid M1 by relaxing a circuit-hyperplane and both M1 and M2 are

Given an edge colouring of a graph with a set of m colours, we say that the graph is m-coloured if each of the m colours is used. For an m-colouring Δ of N(2), the complete graph on N, we denote by FΔ the set all values γ for which there exists an infinite subset X⊂N such that X(2) is γ-coloured. Properties of this set were first studied by Erickson in 1994. Here, we are interested in estimating the

A simple topological graph is k-quasiplanar (k≥2) if it contains no k pairwise crossing edges, and k-planar if no edge is crossed more than k times. In this paper, we explore the relationship between k-planarity and k-quasiplanarity to show that, for k≥2, every k-planar simple topological graph can be transformed into a (k+1)-quasiplanar simple topological graph.

The linear arboricity of a graph G, denoted by la(G), is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in G whose union covers all the edges of G. A famous conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that la(G)≤⌈(Δ(G)+1)/2⌉, where Δ(G) denotes the maximum degree of G. This conjectured upper bound would be best possible

A class of graphs is χ-bounded if there is a function f such that χ(G)≤f(ω(G)) for every induced subgraph G of every graph in the class, where χ,ω denote the chromatic number and clique number of G respectively. In 1987, Gyárfás conjectured that for every c, if C is a class of graphs such that χ(G)≤ω(G)+c for every induced subgraph G of every graph in the class, then the class of complements of members

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced subgraph

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a vertex that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor

Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence