We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group. The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree

In 1943, Hadwiger conjectured that every graph with no Kt+1 minor is t-colorable for every t≥0. While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980 s, Kostochka and Thomason independently proved that every graph with no Kt minor has average degree O(tlog⁡t) and thus is O(tlog⁡t)-list-colorable. Recently, the authors and Song proved

This paper gives a solution of Problem 1.8 in [16]. As a corollary, it is shown that every connected non-bipartite Cayley graph on a solvable group of valency at least three is at most 2-arc-transitive.

The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2n/17700000. On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is <215n/log2⁡(n).

Let G=(V,E) be a graph. A proper total weighting of G is a mapping w:V∪E⟶R such that the following sum for each v∈V:w(v)+∑e∈E(v)w(e) gives a proper vertex colouring of G. For any a,b∈N+, we say that G is total weight (a,b)-choosable if for any {Sv:v∈V}⊂[R]a and {Se:v∈E}⊂[R]b, there exists a proper total weighting w of G such that w(v)∈Sv for v∈V and w(e)∈Se for e∈E. A strengthening of the 1-2-3 Conjecture

We present two extremal results on 4-cycles. Let q be a large even integer. First we prove that every (q2+q+1)-vertex C4-free graph with more than 12q(q+1)2−0.2q edges must be a spanning subgraph of a unique polarity graph. This implies a stability refinement of a special case of the seminal work of Füredi on the extremal number of C4. Second we prove that every (q2+q+1)-vertex graph with 12q(q+1)2+1

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný et al. (2016) [11]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long

We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at −1 up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop “acyclic orientation” analogues of theorems concerning the

The convex dimension of a k-uniform hypergraph is the smallest dimension d for which there is an injective mapping of its vertices into Rd such that the set of k-barycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete k-uniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We

We show that if a graph admits a packing and a covering both consisting of λ many spanning trees, where λ is some infinite cardinal, then the graph also admits a decomposition into λ many spanning trees. For finite λ the analogous question remains open, however, a slightly weaker statement is proved.

We generalize the Harary-Sachs theorem to k-uniform hypergraphs: the codegree-d coefficient of the characteristic polynomial of a uniform hypergraph H can be expressed as a weighted sum of subgraph counts over certain multi-hypergraphs with d edges. We include a detailed description of the aforementioned multi-hypergraphs and a formula for their corresponding weights.

Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most ⌊(n−1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1], Dean (1986) [7], and Bollobás and Scott (1996) [2] it was analogously conjectured that every directed Eulerian graph can be decomposed

The extremal number ex(n,F) of a graph F is the maximum number of edges in an n-vertex graph not containing F as a subgraph. A real number r∈[1,2] is realisable if there exists a graph F with ex(n,F)=Θ(nr). Several decades ago, Erdős and Simonovits conjectured that every rational number in [1,2] is realisable. Despite decades of effort, the only known realisable numbers are 0,1,75,2, and the numbers

An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut∘M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut∘M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified

Let F be a family of r-uniform hypergraphs. The feasible region Ω(F) of F is the set of points (x,y) in the unit square such that there exists a sequence of F-free r-uniform hypergraphs whose shadow density approaches x and whose edge density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x,y)∈Ω(F) is the Turán density π(F), and

A classical result by Hajnal and Szemerédi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a Kr-factor. Namely, any graph on n vertices, with minimum degree δ(G)≥(1−1r)n and r dividing n has a Kr-factor. This result is tight but the extremal examples are unique in that they all have a large independent set which is the bottleneck. Nenadov and Pehova

In 1990, Thomas proved that every graph admits a tree decomposition of minimum width that additionally satisfies a certain vertex-connectivity condition called leanness. This result had many uses and has been extended to several other decompositions. In this paper, we consider tree-cut decompositions, that have been introduced by Wollan (2015) as a possible edge-version of tree decompositions. We show

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian

In the present paper, we introduce the notion of graph functionality, which generalises simultaneously several other graph parameters, such as degeneracy or clique-width, in the sense that bounded degeneracy or bounded clique-width imply bounded functionality. Moreover, we show that this generalisation is proper by revealing classes of graphs of unbounded degeneracy and clique-width, where functionality

The reconfiguration graph Rk(G) for the k-colorings of a graph G has as vertex set the set of all possible k-colorings of G and two colorings are adjacent if they differ in the color of exactly one vertex of G. Let d,k≥1 be integers such that k≥d+1. We prove that for every ϵ>0 and every graph G with n vertices and maximum average degree d−ϵ, Rk(G) has diameter O(n(log⁡n)d−1). This significantly strengthens

Let fr(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, in which the union of any e distinct edges contains at least v+1 vertices. The study of fr(n,v,e) was initiated by Brown, Erdős and Sós more than forty years ago. In the literature, the following conjecture is well known. Conjecture: nk−o(1)k≥2 and e≥3 as n→∞. For r=3,e=3,k=2, the bound n2−o(1)

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have been shown to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved—like, for example, dismantlability—and

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs (Γ,Σ) is stable if Aut(Γ×Σ)≅Aut(Γ)×Aut(Σ) and unstable otherwise, where Γ×Σ is the direct product of Γ and Σ. An unstable graph pair (Γ,Σ) is said to be a nontrivially unstable graph pair if Γ and Σ are connected coprime graphs, at least

Grötzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any

In this paper, we introduce the language of a configuration and of t-point counts for distance-regular graphs (DRGs). Every t-point count can be written as a sum of (t−1)-point counts. This leads to a system of linear equations and inequalities for the t-point counts in terms of the intersection numbers, i.e., a linear constraint satisfaction problem (CSP). This language is a very useful tool for a

In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers n,k,r,s with n≥rk and 2≤s≤r, the r-uniform general Kneser hypergraph KGsr(n,k), has all k-subsets of {1,…,n} as the vertex set and all multi-sets {A1,…,Ar} of k-subsets with s-wise empty intersections as the edge set. The case r=s=2, was considered by Kneser

The prism over a graph G is the Cartesian product of G with the complete graph K2. A graph G is hamiltonian if there exists a spanning cycle in G, and G is prism-hamiltonian if the prism over G is hamiltonian. Rosenfeld and Barnette (1973) [15] conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.

We prove that, for each circle graph H, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to H.

Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the eventual extremal functions for these classes. One of the main tools for obtaining these results is the notion of

We show that for every positive integer k, any tournament with minimum out-degree at least (2+o(1))k2 contains a subdivision of the complete directed graph on k vertices, where each path of the subdivision has length at most 3. This result is best possible on the minimum out-degree condition (up to a multiplicative factor of 8), and it is tight with respect to the length of the paths. It may be viewed

We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function f:Z+→Z+, such that for every integer g>0, every graph of treewidth at least f(g) contains the (g×g)-grid as a minor. For every integer g>0, let f(g) be the smallest value for which the theorem

In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of

Hedetniemi conjectured in 1966 that χ(G×H)=min⁡{χ(G),χ(H)} for all graphs G and H. Here G×H is the graph with vertex set V(G)×V(H) defined by putting (x,y) and (x′,y′) adjacent if and only if xx′∈E(G) and yy′∈E(H). This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let p be the minimum number of vertices in a graph of odd girth 7 and fractional

For a graph H, letc∞(H)=limn→∞⁡max⁡|E(G)|n, where the maximum is taken over all graphs G on n vertices not containing H as a minor. Thus c∞(H) is the asymptotic maximum density of graphs not containing H as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on c∞(H) for disconnected graphs H. In particular, we determine c∞(H) whenever H is a union of cycles. Finally, we

There is error in the proof of Theorem 1.1, Subcase 2.1 of [J. Combin. Theory Ser. B 81 (1) (2001) 77–99]. Here, we fix that error.

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally

A path-matching of order p is a vertex disjoint union of nontrivial paths spanning p vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers r,p1,…,pr, define RPM(p1,…,pr) to be the smallest integer n such that in any r-coloring of the edges of Kn there exists

Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that∑fface ofG(|f|−4)≤κ(g+t+c−1) for a fixed constant κ, thus generalizing and strengthening several known results. As a corollary, we prove that every triangle-free graph G embedded in a surface of genus g contains a set of O(g) vertices such

A classical result of Erdős, Gyárfás and Pyber states that any r-edge-coloured complete graph has a partition into O(r2log⁡r) monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant c such that any r-edge-coloured graph on n vertices with minimum degree at least n/2+c⋅rlog⁡n has a partition into O(r2) monochromatic

For a 2-connected graph G on n vertices and two vertices x,y∈V(G), we prove that there is an (x,y)-path of length at least k, if there are at least n−12 vertices in V(G)\{x,y} of degree at least k. This strengthens a celebrated 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

We prove that, for every natural number k, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in [k,2k+9]. We also show that this bound is close to being optimal by constructing, for every even k≥4, an infinite family of 3-connected cubic planar graphs that contain no cycle whose length is in [k,2k+1].

The interval number of a graph G is the minimum k such that one can assign to each vertex of G a union of k intervals on the real line, such that G is the intersection graph of these sets, i.e., two vertices are adjacent in G if and only if the corresponding sets of intervals have non-empty intersection. Scheinerman and West (1983) [14] proved that the interval number of any planar graph is at most

Let M be a 3-connected matroid, and let N be a 3-connected minor of M. 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 second in a series of three papers where we describe the structures that arise when it is not possible to find an N-detachable pair in M. In the first paper in the series, we showed that, under mild assumptions

We show that the tangle space of a graph, which compactifies it, is a quotient of its Stone-Čech remainder obtained by contracting the connected components.

We show that the number of Laplacian eigenvalues less than the average degree 2−2n of a tree having n vertices is at least ⌈n2⌉.

In the study of graph edge coloring for simple graphs, a graph G is called Δ-critical if Δ(G)=Δ, χ′(G)=Δ(G)+1 and χ′(H)<χ′(G) for every proper subgraph H of G. In this paper, we prove a new adjacency result of critical graphs which allows us to control the degree of vertices with distance four. Combining this result with a previous theorem proved by the authors, we show that for every ϵ>0, if G is

We show that a graph G has a normal spanning tree if and only if its vertex set is the union of countably many sets each separated from any subdivided infinite clique in G by a finite set of vertices. This proves a conjecture by Brochet and Diestel from 1994, giving a common strengthening of two classical normal spanning tree criterions due to Jung and Halin. Moreover, our method gives a new, algorithmic

We find Dirac-type sufficient conditions for a hypergraph H with few edges to be hamiltonian. We also show that these conditions guarantee that H is super-pancyclic, i.e., for each A⊆V(H) with |A|≥3, H contains a Berge cycle with vertex set A. We mostly use the language of bipartite graphs, because every bipartite graph is the incidence graph of a multihypergraph. In particular, we extend some results

We prove that for any positive integers k and d, if a graph G has maximum average degree at most 2k+2dd+k+1, then G decomposes into k+1 pseudoforests C1,…,Ck+1 so that for at least one of the pseudoforests, each connected component has at most d edges.

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node 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 a wheel

Let Λ be an infinite connected graph, and let v0 be a vertex of Λ. We consider the following positional game. Two players, Maker and Breaker, play in alternating turns. Initially all edges of Λ are marked as unsafe. On each of her turns, Maker marks p unsafe edges as safe, while on each of his turns Breaker takes q unsafe edges and deletes them from the graph. Breaker wins if at any time in the game

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.

Given a positive integer s, a graph G is s-Ramsey for a graph H, denoted G→(H)s, if every s-colouring of the edges of G contains a monochromatic copy of H. The s-colour size-Ramsey number rˆs(H) of a graph H is defined to be rˆs(H)=min⁡{|E(G)|:G→(H)s}. We prove that, for all positive integers k and s, we have rˆs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn.

Fix d≥3. We show the existence of a constant c>0 such that any graph of diameter at most d has average distance at most d−cd3/2n, where n is the number of vertices. Moreover, we exhibit graphs certifying sharpness of this bound up to the choice of c. This constitutes an asymptotic solution to a longstanding open problem of Plesník. Furthermore we solve the problem exactly for digraphs if the order

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in R3 is not greater than 13.92. We also find new upper bounds for the average degree of contact graphs in R4 and R5.

This paper consists of two major parts. In the first part, the relations between Tutte orientations and circular flows are explored. Tutte orientation (modulo orientation) was first observed by Tutte for the study of 3-flow problem, and later extended by Jaeger for circular (2+1/p)-flows. In this paper, it is extended for circular λ-flows for all rational numbers λ. This theorem is one of the key tools

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism

We show that there are planar graphs that require four pages in any book embedding.

Given a partial Steiner triple system (STS) of order n, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order n with at most r≤εn2 triples, it can always be embedded into a complete STS of order n+O(r), which is asymptotically optimal

Given a graph H and a set of graphs F, let ex(n,H,F) denote the maximum possible number of copies of H in an F-free graph on n vertices. We investigate the function ex(n,H,F), when H and members of F are cycles. Let Ck denote the cycle of length k and let Ck={C3,C4,…,Ck}. We highlight the main results below. (i) We show that ex(n,C2l,C2k)=Θ(nl) for any l,k≥2. Moreover, in some cases we determine it