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Treewidth and Hadwiger number are two of the most important parameters in structural graph theory. This paper studies graph classes in which large treewidth implies the existence of a large complete graph minor. To formalise this, we say that a graph class $\mathcal{G}$ is (tw,had)-bounded if there is a function $f$ (called the (tw,had)-bounding function) such that tw$(G)$ $\leq$ $f$(had$(G)$) for every graph $G \in \mathcal{G}$. We characterise (tw,had)-bounded graph classes as those that exclude some planar graph as an induced minor, and use this characterisation to show that every proper vertex-minor-closed class is (tw,had)-bounded. Furthermore, we demonstrate that any (tw,had)-bounded graph class has a (tw,had)-bounding function in O(had$(G)^9$polylog(had$(G)$)). Our bound comes from the bound for the Grid Minor Theorem given by Chuzhoy and Tan, and any quantitative improvement to their result will lead directly to an improvement to our result. More strongly, we conjecture that every (tw,had)-bounded graph class has a linear (tw,had)-bounding function. In support of this conjecture, we show that it holds for the class of outer-string graphs, and for a natural generalisation of outer-string graphs: intersection graphs of strings rooted at the boundary of a fixed surface. We also verify our conjecture for low-rank perturbations of circle graphs, which is an important step towards verifying it for all proper vertex-minor-closed classes., Comment: 26 pages, 6 Figures

In 1972, Kotzig proved that for every even $n$, the complete graph $K_n$ can be decomposed into $\lceil\log_2n\rceil$ edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite decompositions of $(n,d,\lambda)$-graphs, where $n$ is an even integer and $d_0\leq d\leq n-1$ for some absolute constant $d_0$. With a randomized algorithm, we prove that such an $(n,d,\lambda)$-graph with $\lambda\leq d/12$ can be decomposed into at most $\log_2 d + 36$ regular bipartite spanning subgraphs. This is best possible up to the additive constant term. As a consequence, we also improve the best known bounds on $\lambda = \lambda(d)$ by Ferber and Jain (2020) to guarantee that an $(n,d,\lambda)$-graph on an even number of vertices admits a $1$-factorization, showing that $\lambda \leq cd$ is sufficient for some absolute constant $c > 0$., Comment: 23 pages, 1 figure

Motivated by colouring minimal Cayley graphs, in 1978, Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge-colouring in which each cycle contains no colour exactly once., Comment: 11 pages, 3 figures

We present progress on two old conjectures about longest cycles in graphs. The first conjecture, due to Thomassen from 1978, states that apart from a finite number of exceptions, all connected vertex-transitive graphs contain a Hamiltonian cycle. The second conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards both of the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle of length at least $\Omega(n^{13/21})$, improving on $\Omega(n^{3/5})$ from [De Vos, arXiv:2302:04255, 2023]. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $\Omega(r^{5/8})$ vertices, improving on $\Omega(r^{3/5})$ from [Chen, Faudree and Gould, J. Combin. Theory, Ser.~ B, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming., Comment: 13 pages, 3 figures. Code available at https://github.com/tjeremie/Long-cycles

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $\mathcal{F}$ is a family in $\mathcal{B}_n$ with $|\mathcal{F}|\ge (q-1+\varepsilon){n\choose \lfloor n/2\rfloor}$ for some $q\ge k$, then $\mathcal{F}$ contains on the order of as many induced copies of $P$ as is contained in the $q$ middle layers of the Boolean lattice. This generalizes results of Bukh and Boehnlein and Jiang which guaranteed a single such copy in non-induced and induced settings respectively. * The number of induced $P$-free families of $\mathcal{B}_n$ is $2^{(k-1+o(1)){n\choose \lfloor n/2\rfloor}}$, strengthening recent independent work of Balogh, Garcia, Wigal who obtained the same bounds in the non-induced setting. * The largest induced $P$-free subset of a $p$-random subset of $\mathcal{B}_n$ for $p\gg n^{-1}$ has size at most $(k-1+o(1))p{n\choose \lfloor n/2\rfloor}$, generalizing previous work of Balogh, Mycroft, and Treglown and of Collares and Morris for the case when $P$ is a chain. All three results are asymptotically tight and give affirmative answers to general conjectures of Gerbner, Nagy, Patk\'os, and Vizer in the case of tree posets., Comment: 37 pages; We significantly extend the previous version, which addressed supersaturation of tree posets, to also obtain tight enumeration results for tree poset free subsets of the Boolean lattice and tight upper bounds for the largest tree poset free subset of random subsets of the Boolean lattice

For $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is $o(q^n)$ as $n \to \infty$. By counting affine homomorphisms between subsets of $\mathbb F_q^n$, we derive new bounds and give new proofs of some previously known bounds for certain affine extremal numbers. At the same time, we establish corresponding supersaturation results. We connect these bounds to certain Ramsey-type numbers in vector spaces over finite fields. For $s,t \geq 1$, let $R_q(s,t)$ denote the minimum $n$ such that in every red-blue coloring of the one-dimensional subspaces of $\mathbb F_q^n$, there is either a red $s$-dimensional subspace or a blue $t$-dimensional subspace of $\mathbb F_q^n$. The existence of these numbers is a special case of a well-known theorem of Graham, Leeb, Rothschild. We improve the best known upper bounds on $R_2(2,t)$, $R_3(2,t)$, $R_2(t,t)$, and $R_3(t,t)$., Comment: 19 pages, 1 figure

We show that Erd\H{o}s-R\'enyi random graphs $G(n,p)$ with constant density $p<1$ have correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger's conjecture for correspondence coloring. The proof follows from a simple sufficient condition for correspondence colorability in terms of the numbers of independent sets.

Given two $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm{ex}(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm{ex}(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine the order of magnitude of $\rm{ex}\left(G_{n,p}^{(r)}, C_{2\ell}^{(r)}\right)$ for all $r\geq 4$ and all $\ell\geq 2$ up to polylogarithmic factors for all values of $p=p(n)$. Our proof is based on the container method and uses a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:2007.10320

A common problem in graph colouring seeks to decompose the edge set of a given graph into few similar and simple subgraphs, under certain divisibility conditions. In 1987 Wormald conjectured that the edges of every cubic graph on $4n$ vertices can be partitioned into two isomorphic linear forests. We prove this conjecture for large connected cubic graphs. Our proof uses a wide range of probabilistic tools in conjunction with intricate structural analysis, and introduces a variety of local recolouring techniques., Comment: 49 pages, many figures

We show that any connected Cayley graph $\Gamma$ on an Abelian group of order $2n$ and degree $\tilde{\Omega}(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $\Gamma$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Pl\"{u}nnecke-Rusza-Petridis inequality from additive combinatorics., Comment: 21 pages, fixed minor typos and citations

We show that for every integer $m \ge 2$ and large $n$, every properly edge-coloured graph on $n$ vertices with at least $n (\log n)^{53}$ edges contains a rainbow subdivision of $K_m$. This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs., Comment: 22 pages, journal version

We show that for any integer $t\geq 2$, every properly edge-coloured graph on $n$ vertices with more than $n^{1+o(1)}$ edges contains a rainbow subdivision of $K_t$. Note that this bound on the number of edges is sharp up to the $o(1)$ error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Tur\'an numbers. Our method relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to find robust expanders in the coloured setting., Comment: 8 pages, journal version

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and R\"odl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends and strongly generalises recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs., Comment: 32 pages (41 including appendix), 6 figures

Given $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $\rm ex(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $\ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.

A well-known result of Verstra\"ete \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of $G$. We establish two extensions of Verstra\"ete's result for linear cycles in linear $r$-uniform hypergraphs. We show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_1=c_1(r)$ and $c_2=c_2(r,k)$, such that every linear $r$-uniform hypergraph $G$ with average degree $d(G)\geq c_1 k$ contains linear cycles of $k$ consecutive even lengths, the shortest of which is at most $2\lceil \frac{ \log n}{\log (d(G)/k)-c_2}\rceil$. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Tur\'an number of $C^r_{2k}$ with improved coefficients. Furthermore, we show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_3=c_3(r)$ and $c_4=c_4(r)$ such that every $n$-vertex linear $r$-uniform graph with average degree $d(G)\geq c_3k$, contains linear cycles of $k$ consecutive lengths, the shortest of which has length at most $6\lceil \frac{\log n}{\log (d(G)/k)-c_4} \rceil +6$. Both the degree condition and the shortest length among the cycles guaranteed are best possible up to a constant factor.

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 $n-O(\log{n}/\log{\log{n}})$, improving the celebrated bound of $n-O(\log^2n)$ by Hatami and Shor. Our approach (different from that of Hatami-Shor) is quite general and gives several other applications as well. We obtain a new lower bound on a 40 year old conjecture of Brouwer on the maximum matching in Steiner triple systems, showing that every such system of order $n$ is guaranteed to have a matching of size $n/3-O(\log{n}/\log{\log{n}})$. This substantially improves the current best result of Alon, Kim and Spencer which has the error term of order $n^{1/2+o(1)}$. Finally, we also show that $O(n\log{n}/\log{\log{n}})$ many symbols in Latin arrays suffice to guarantee a full transversal, improving on previously known bound of $n^{2-\varepsilon}$. The proofs combine in a novel way the semirandom method together with the robust expansion properties of edge coloured pseudorandom graphs to show the existence of a rainbow matching covering all but $O(\log n/\log{\log{n}})$ vertices. All previous results, based on the semi-random method, left uncovered at least $\Omega(n^{\alpha})$ (for some constant $\alpha$) vertices.

We show that if a multigraph $G$ with maximum edge-multiplicity of at most $\frac{\sqrt{n}}{\log^2 n}$, is edge-coloured by $n$ colours such that each colour class is a disjoint union of cliques with at least $2n + o(n)$ vertices, then it has a full rainbow matching, that is, a matching where each colour appears exactly once. This asymptotically solves a question raised by Clemens, Ehrenm\"uller and Pokrovskiy, and is related to problems on algebras of sets studied by Grinblat in [Grinblat 2002].

Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that every $K_5$-minor-free graph has tree-chromatic number at most $4$, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path-chromatic number., Comment: 11 pages, 0 figures

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and Sudakov, solving a conjecture of Erd\H{o}s, Faudree and S\'os, shows that any $C$-Ramsey $N$-vertex graph contains an induced subgraph with $\Omega_C(N^{1/2})$ distinct degrees. We improve this to $\Omega_C(N^{2/3})$, which is tight up to the constant factor. We also show that any $N$-vertex graph with $N > (k-1)(n-1)$ and $n\geq n_0(k) = \Omega (k^9)$ either contains a homogeneous set of order $n$ or an induced subgraph with $k$ distinct degrees. The lower bound on $N$ here is sharp, as shown by an appropriate Tur\'an graph, and confirms a conjecture of Narayanan and Tomon., Comment: 13 pages

The celebrated Hadwiger's conjecture states that if a graph contains no $K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set of size at least $n/t$. In 1982, Duchet and Meyniel proved that this bound holds within a factor $2$. Their bound has been improved; most notably in an absolute factor by Fox, which was later improved by Balogh and Kostochka. Here we consider the same question for triangle-free graphs. By the results of Shearer and Kostochka and Thomason, it follows that any triangle-free graph with no $K_t$ minor has an independent set of size $\Omega(\tfrac{\sqrt{\log{t}}}{t}n)$. We show that a much larger independent set exists; for all sufficiently large $t$ every triangle-free graph on $n$ vertices with no $K_t$-minor has an independent set of size $ \tfrac{n}{t^{1-\varepsilon}}$. This answers a question of Sergey Norin.

A graph $G$ is Ramsey for a graph $H$ if every 2-colouring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a `sparse' graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree., Comment: Following the release of our original preprint, a positive answer to our Question 5.2 has been given in arXiv:1907.03466 . Also an alternative proof of Theorem 1.3 has been provided by Jonathan Rollin, which we added to the preprint. Subsequently concluding remarks have been changed, the remaining changes are minor

A long-standing conjecture of Erd\H{o}s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=\Theta(n^r)$. So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$, for integers $k\geq 2$. In this paper we add a new form of rationals for which the conjecture is true; $2-2/(2k+1)$, for $k\geq 2$. This in its turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erd\H{o}s and Simonovits's conjecture where one replaces a single graph by a family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erd\H{o}s and Simonovits's conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon's conjecture. We also prove the an upper bound on the Tur\'an's number of $\theta$-graphs in an asymmetric setting and employ this result to obtain yet another new rational exponent for Tur\'an exponents; $r=7/5$.

Akbari and Alipour conjectured that any Latin array of order $n$ with at least $n^2/2$ symbols contains a transversal. We confirm this conjecture for large $n$, and moreover, we show that $n^{399/200}$ symbols suffice., Comment: 6 pages, 1 figure

For any $r$-graph $H$, we consider the problem of finding a rainbow $H$-factor in an $r$-graph $G$ with large minimum $\ell$-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an $H$-factor., Comment: 10 pages

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities $o(n)$ that is properly edge-coloured by $n$ colours with at least $n + o(n)$ edges of each colour there must be a matching of size $n-O(1)$ that uses each colour at most once., Comment: 7 pages

A classic result of Erd\H{o}s and, independently, of Bondy and Simonovits says that the maximum number of edges in an $n$-vertex graph not containing $C_{2k}$, the cycle of length $2k$, is $O( n^{1+1/k})$. Simonovits established a corresponding supersaturation result for $C_{2k}$'s, showing that there exist positive constants $C,c$ depending only on $k$ such that every $n$-vertex graph $G$ with $e(G)\geq Cn^{1+1/k}$ contains at least $c\left(\frac{e(G)}{v(G)}\right)^{2k}$ many copies of $C_{2k}$, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper, we extend Simonovits' result to a supersaturation result of $r$-uniform linear cycles of even length in $r$-uniform linear hypergraphs. Our proof is self-contained and includes the $r=2$ case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine the Tur\'an number of the extension of an $r$-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Tur\'an number for $r=3$. As the key ingredient of the proof we show that the Lagrangian of intersecting $r$-graphs is maximized by principally intersecting $r$-graphs for $r \geq 4$.

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} are internally disjoint from phi(V(H)). We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al., Comment: 12 pages, 1 figure

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $r-2$ new vertices. Keevash and Sidorenko~ have previously determined Tur\'an densities of two families of hypergraph extensions. We determine the Tur\'an numbers for these families, using classical stability techniques and new tools introduced in our earlier paper.

For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying conjectures of Reed and Wood, and Harvey and Wood. We derive the above result from a theorem which allows us to find two vertex disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.

We show that every planar, 4-connected, K2;5-minor- free graph is the square of a cycle of even length at least six., Comment: 6 pages

The family $\Sigma_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $|D_1\cap D_2|=r-1$ and $D_1 \triangle D_2 \subseteq D_3$. A generalized triangle, $\mathcal{T}_r \in \Sigma_r$ is an $r$-graph on $\{1,2,\ldots,2r-1\}$ with three edges $D_1, D_2, D_3$, such that $D_1=\{1,2,\dots,r-1, r\}, D_2= \{1, 2, \dots, r-1, r+1 \}$ and $D_3 = \{r, r+1, \dots, 2r-1\}.$ Frankl and F\"{u}redi conjectured that for all $r\geq 4$, $ex(n,\Sigma_r) = ex(n,\mathcal{T}_r )$ for all sufficiently large $n$ and they also proved it for $r=3$. Later, Pikhurko showed that the conjecture holds for $r=4$. In this paper we determine $ex(n,\mathcal{T}_5)$ and $ex(n,\mathcal{T}_6)$ for sufficiently large $n$, proving the conjecture for $r=5,6$., Comment: 31 pages

Erd\H{o}s conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\frac{5}{14}n$ and for graphs with average degree at least $(\frac{2}{5} - \varepsilon)n$ for some absolute $\varepsilon >0$. Moreover, we show that the conjecture is true for graphs which are close to the Petersen graph in edit distance., Comment: 23 pages

An interval t-coloring of a graph G is a proper edge-coloring of G with colors 1,2,...,t such that at least one edge of G is colored by i, i=1,2,...,t, and the edges incident to each vertex v\in V(G) are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge-colorings of various graph products are investigated., Comment: 4 pages