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Let $ a,b \in {\bf Z}^+$ $r=a + b$, and let $T$ be a tree with parts $U = \{u_1,u_2,\dots,u_s\}$ and $V = \{v_1,v_2,\dots,v_t\}$. Let $U_1, \dots ,U_s$ and $V_1, \dots, V_t$ be disjoint sets, such that {$|U_i|=a$ and $|V_j|=b$ for all $i,j$}. The {\em $(a,b)$-blowup} of $T$ is the $r$-uniform hypergraph with edge set $ {\{U_i \cup V_j : u_iv_j \in E(T)\}.}$ We use the $\Delta$-systems method to prove the following Tur\' an-type result. Suppose $a,b,t\in {\bf Z}^+$, $r=a+b\geq 3$,{ $a\geq 2$,} and $T$ is a fixed tree of diameter $4$ in which the degree of the center vertex is $t$. Then there exists a $C=C(r,t,T)>0$ such that $ |\mathcal{H}|\leq (t-1){n\choose r-1} +Cn^{r-2}$ for every $n$-vertex $r$-uniform hypergraph $\mathcal{H}$ {not containing an $(a,b)$-blowup of $T$}. This is {asymptotically exact} when $t\leq|V(T)|/2$. A stability result is also presented.

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of (unlabeled) subgraphs obtained from it by deleting $\ell$ vertices. An $n$-vertex graph is $\ell$-reconstructible if it is determined by its $(n-\ell)$-deck, meaning that no other graph has the same deck. We prove that every tree with at least $6\ell+11$ vertices is $\ell$-reconstructible.

An interesting generalization of list coloring is so called DP-coloring (named after Dvo\v{r}\'ak and Postle). We study $(i,j)$-defective DP-colorings of simple graphs. Define $g_{DP}(i,j,n)$ to be the minimum number of edges in an $n$-vertex DP-$(i,j)$-critical graph. We prove sharp bounds on $g_{DP}(i,j,n)$ for $i=1,2$ and $j\geq 2i$ for infinitely many $n$., Comment: 19 pages, 1 figure. arXiv admin note: text overlap with arXiv:2006.10244

The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree $\Delta\geq 3$ that is not the complete graph $K_{\Delta+1}$ or the complete bipartite graph $K_{\Delta,\Delta}$ admits an equitable coloring with $\Delta$ colors. For planar graphs, the conjecture has been confirmed for $\Delta\geq 13$ by Yap and Zhang and for $9\leq \Delta\leq 12$ by Nakprasit. In this paper, we present a proof that confirms the conjecture for graphs embeddable into a surface with non-negative Euler characteristic with maximum degree $\Delta\geq 9$ and for planar graphs with maximum degree $\Delta\geq 8$., Comment: 20 pages, 2 figures

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally semicomplete digraphs. In this note, we state a similar conjecture for all digraphs, show that the two conjectures are equivalent, and prove that both conjectures hold for a class of digraphs containing all orientations of 4-colorable graphs (in particular, of all planar graphs).

We study the minimum number of maximum matchings in a bipartite multigraph G with parts $X$ and $Y$ under various conditions, refining the well-known lower bound due to M. Hall. When $|X|=n$, every vertex in $X$ has degree at least $k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum is $r!(k-r+1)$ when $n\ge r$ and is $[r+n(k-r)]\prod_{i=1}^{n-1}(r-i)$ when $n

A hypergraph $H$ is hamiltonian-connected if for any distinct vertices $x$ and $y$, $H$ contains a hamiltonian Berge path from $x$ to $y$. We find for all $3\leq r, Comment: 22 pages, 1 figure. Final version to appear in the European Journal of Combinatorics

We refine a property of $2$-connected graphs described in the classical paper of Dirac from 1952 and use the refined property to somewhat shorten Dirac's proof of the fact that each $2$-connected $n$-vertex graph with minimum degree at least $k$ has a cycle of length at least $\min\{n,2k\}$.

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We prove a hypergraph version of this result: for $n \geq k \geq r+2 \geq 5$, every $2$-connected $r$-uniform $n$-vertex hypergraph with minimum degree at least ${k-1 \choose r-1} + 1$ has a Berge cycle of length at least $\min\{2k, n\}$. The bound is exact for all $k\geq r+2\geq 5$., Comment: 22 pages, 1 figure

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, Comment: 23 pages. Updated statement of Claim 3.1

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs having no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). It is known that this fails when $n=2\ell$., Comment: 16 pages

For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $n\to\infty$.

For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$ is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that, for each $1\leq i \leq k$, the distance between any two distinct $x,y\in V_i$ is at least $s_i + 1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs $G$ with arbitrarily large $\chi_p(G)$. Recently, there was a series of papers on packing $(s_1, s_2, \ldots, s_k)$-colorings of subcubic graphs in various classes. We show that every $2$-connected subcubic outerplanar graph has a packing $(1,1,2)$-coloring and every subcubic outerplanar graph is packing $(1,1,2,4)$-colorable. Our results are sharp in the sense that there are $2$-connected subcubic outerplanar graphs that are not packing $(1,1,3)$-colorable and there are subcubic outerplanar graphs that are not packing $(1,1,2,5)$-colorable. We also show subcubic outerplanar graphs that are not packing $(1,2,2,4)$-colorable and not packing $(1,1,3,4)$-colorable., Comment: 13 pages, 5 figures

A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that $3$-regular graphs are $2$-reconstructible.

A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following \L uczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings., Comment: 29 pages, 2 figures

The $k$-deck of a graph is the multiset of its subgraphs induced by $k$ vertices. A graph or graph property is $l$-reconstructible if it is determined by the deck of subgraphs obtained by deleting $l$ vertices. We show that the degree list of an $n$-vertex graph is $3$-reconstructible when $n\ge7$, and the threshold on $n$ is sharp. Using this result, we show that when $n\ge7$ the $(n-3)$-deck also determines whether an $n$-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are $2$-reconstructible when $n\ge6$, which are also sharp., 12 pages

A graph $G$ arrows a graph $H$ if in every $2$-edge-coloring of $G$ there exists a monochromatic copy of $H$. Schelp had the idea that if the complete graph $K_n$ arrows a small graph $H$, then every "dense" subgraph of $K_n$ also arrows $H$, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large $n$, if $n = 3t+r$ where $r \in \{0,1,2\}$ and $G$ is an $n$-vertex graph with $\delta(G) \ge (3n-1)/4$, then for every $2$-edge-coloring of $G$, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same color, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the same color. Our result is tight in the sense that no longer cycles (of length $>2t+r$) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large $n$, every $(3t-1)$-vertex graph $G$ with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with $2n$ vertices. Moreover, it implies for sufficiently large $n$ the conjecture by Benevides, {\L}uczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every $n$-vertex graph $G$ with $\delta(G) \ge 3n/4$, in each $2$-edge-coloring of $G$ there exists a monochromatic cycle of length at least $2t+r$., Comment: 16 pages

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.

We find Dirac-type sufficient conditions for a hypergraph $\mathcal H$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal H$ is {\em super-pancyclic}, i.e., for each $A \subseteq V(\mathcal H)$ with $|A| \geq 3$, $\mathcal H$ 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 of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high minimum degree. Furthermore, we prove a conjecture of Jackson from 1981 on long cycles in 2-connected bipartite graphs.

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and discover a general partitioning phenomenon which allows us to determine the order of magnitude of the extremal function for various ordered and convex geometric hypergraphs. A special case is the ordered $n$-vertex $r$-graph $F$ consisting of two disjoint sets $e$ and $f$ whose vertices alternate in the ordering. We show that for all $n \geq 2r + 1$, the maximum number of edges in an ordered $n$-vertex $r$-graph not containing $F$ is exactly \[ {n \choose r} - {n - r \choose r}.\] This could be considered as an ordered version of the Erd\H{o}s-Ko-Rado Theorem, and generalizes earlier results of Capoyleas and Pach and Aronov-Dujmovi\v{c}-Morin-Ooms-da Silveira., Comment: 16 pages, 2 figures

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The questions on the value of the maximum of $\chi_p(G)$ and of $\chi_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $\chi_p(D(G))\leq 5$ for any subcubic $G$, and later Bresar, Klavzar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $\chi_p(G)$ is not bounded in the class of subcubic graphs $G$. In contrast, in this paper we show that $\chi_p(D(G))$ is bounded in this class, and does not exceed $8$., Comment: 20 pages, 15 figures

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvo\v{r}\'{a}k and Postle. In this paper we introduce and study the fractional DP-chromatic number $\chi_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $\chi_{DP}^\ast(G) \leqslant 2$: they are precisely the graphs with no odd cycles and at most one even cycle. By a theorem of Alon, Tuza, and Voigt, the fractional list-chromatic number $\chi_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $\chi^\ast(G)$. This equality does not extend to fractional DP-colorings. Moreover, we show that the difference $\chi^\ast_{DP}(G) - \chi^\ast(G)$ can be arbitrarily large, and, furthermore, $\chi^\ast_{DP}(G) \geq d/(2 \ln d)$ for every graph $G$ of maximum average degree $d \geq 4$. On the other hand, we show that this asymptotic lower bound is tight for a large class of graphs that includes all bipartite graphs as well as many graphs of high girth and high chromatic number., Comment: 13 pages, 1 figure

We show that for each $k\geq 4$ and $n>r\geq k+1$, every $n$-vertex $r$-uniform hypergraph with no Berge cycle of length at least $k$ has at most $\frac{(k-1)(n-1)}{r}$ edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Gy��ri, Katona and Lemons that for $n>r\geq k\geq 3$, every $n$-vertex $r$-uniform hypergraph with no Berge path of length $k$ has at most $\frac{(k-1)n}{r+1}$ edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least $2k$, and then translate the results into the language of multi-hypergraphs.

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ and $g\geq 2k+2$, almost every $n$-vertex cubic graph of girth at least $g$ has the packing chromatic number greater than $k$., Comment: 16 pages, 2 figures

The Erd��s--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erd��s--Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e(G) \leq \max\{h(n,k,2),h(n,k,\lfloor \frac{k-1}{2}\rfloor)\}$, where $h(n,k,a) := {k - a \choose 2} + a(n - k + a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,\lfloor \frac{k-1}{2}\rfloor)$ edges. In this paper, we complete a stability theorem which strengthens Kopylov's result. In particular, we show that for $k \geq 3$ odd and all $n \geq k$, every $n$-vertex $2$-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e(G) \leq \max\{h(n,k,3),h(n,k,\frac{k-3}{2})\}$. The upper bound for $e(G)$ here is tight.

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erd��s proved that when $n \geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree $��(G) \geq d$ has at most $h(n,d)$ edges. He also provides a sharpness example $H_{n,d}$ for all such pairs $n,d$. Previously, we showed a stability version of this result: for $n$ large enough, every nonhamiltonian graph $G$ on $n$ vertices with $��(G) \geq d$ and more than $h(n,d+1)$ edges is a subgraph of $H_{n,d}$. In this paper, we show that not only does the graph $H_{n,d}$ maximize the number of edges among nonhamiltonian graphs with $n$ vertices and minimum degree at least $d$, but in fact it maximizes the number of copies of any fixed graph $F$ when $n$ is sufficiently large in comparison with $d$ and $|F|$. We also show a stronger stability theorem, that is, we classify all nonhamiltonian $n$-graphs with $��(G) \geq d$ and more than $h(n,d+2)$ edges. We show this by proving a more general theorem: we describe all such graphs with more than ${n-(d+2) \choose k} + (d+2){d+2 \choose k-1}$ copies of $K_k$ for any $k$., Edited 04/05/17 to add another reference

Let $k \geq 3$ be an integer, $h_{k}(G)$ be the number of vertices of degree at least $2k$ in a graph $G$, and $\ell_{k}(G)$ be the number of vertices of degree at most $2k-2$ in $G$. Dirac and Erd��s proved in 1963 that if $h_{k}(G) - \ell_{k}(G) \geq k^{2} + 2k - 4$, then $G$ contains $k$ vertex-disjoint cycles. For each $k\geq 2$, they also showed an infinite sequence of graphs $G_k(n)$ with $h_{k}(G_k(n)) - \ell_{k}(G_k(n)) = 2k-1$ such that $G_k(n)$ does not have $k$ disjoint cycles. Recently, the authors proved that, for $k \geq 2$, a bound of $3k$ is sufficient to guarantee the existence of $k$ disjoint cycles and presented for every $k$ a graph $G_0(k)$ with $h_{k}(G_0(k)) - \ell_{k}(G_0(k))=3k-1$ and no $k$ disjoint cycles. The goal of this paper is to refine and sharpen this result: We show that the Dirac-Erd��s construction is optimal in the sense that for every $k \geq 2$, there are only finitely many graphs $G$ with $h_{k}(G) - \ell_{k}(G) \geq 2k$ but no $k$ disjoint cycles. In particular, every graph $G$ with $|V(G)| \geq 19k$ and $h_{k}(G) - \ell_{k}(G) \geq 2k$ contains $k$ disjoint cycles.

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$ and $e(n,d):= \max\{h(n,d),h(n, \left \lfloor \frac{n-1}{2} \right \rfloor)\}$. Because $h(n,d)$ is quadratic in $d$, there exists a $d_0(n)=(n/6)+O(1)$ such that $e(n,1)> e(n, 2)> \dots >e(n,d_0)=e(n, d_0+1)=\dots = e(n,\left \lfloor \frac{n-1}{2} \right \rfloor)$. A theorem by Erd��s states that for $d\leq \left \lfloor \frac{n-1}{2} \right \rfloor$, any $n$-vertex nonhamiltonian graph $G$ with minimum degree $��(G) \geq d$ has at most $e(n,d)$ edges, and for $d > d_0(n)$ the unique sharpness example is simply the graph $K_n-E(K_{\lceil (n+1)/2\rceil})$. Erd��s also presented a sharpness example $H_{n,d}$ for each $1\leq d \leq d_0(n)$. We show that if $d< d_0(n)$ and a $2$-connected, nonhamiltonian $n$-vertex graph $G$ with $��(G) \geq d$ has more than $e(n,d+1)$ edges, then $G$ is a subgraph of $H_{n,d}$. Note that $e(n,d) - e(n, d+1) = n - 3d - 2 \geq n/2$ whenever $d< d_0(n)-1$., Edited 04/05/17 to add a new reference

Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erd��s proved that $G$ contains $k$ (vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \ge k^{2} + 2k - 4$. The main result of this paper is that for $k \ge 2$, every graph $G$ with $|V(G)| \ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)| - |L_{k}(G)| \ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k \ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \ge 3k$, then $G$ contains $k$ disjoint cycles. This generalizes the Corr��di-Hajnal Theorem, which states that every graph $G$ with $H_{k}(G) = V(G)$ and $|H_{k}(G)| \ge 3k$ contains $k$ disjoint cycles., 13 pages

The Erd��s-Gallai Theorem states that for $k \geq 2$, every graph of average degree more than $k - 2$ contains a $k$-vertex path. This result is a consequence of a stronger result of Kopylov: if $k$ is odd, $k=2t+1\geq 5$, $n \geq (5t-3)/2$, and $G$ is an $n$-vertex $2$-connected graph with at least $h(n,k,t) := {k-t \choose 2} + t(n -k+ t)$ edges, then $G$ contains a cycle of length at least $k$ unless $G = H_{n,k,t} := K_n - E(K_{n - t})$. In this paper we prove a stability version of the Erd��s-Gallai Theorem: we show that for all $n \geq 3t > 3$, and $k \in \{2t+1,2t + 2\}$, every $n$-vertex 2-connected graph $G$ with $e(G) > h(n,k,t-1)$ either contains a cycle of length at least $k$ or contains a set of $t$ vertices whose removal gives a star forest. In particular, if $k = 2t + 1 \neq 7$, we show $G \subseteq H_{n,k,t}$. The lower bound $e(G) > h(n,k,t-1)$ in these results is tight and is smaller than Kopylov's bound $h(n,k,t)$ by a term of $n-t-O(1)$., Dedicated to the memory of G. N. Kopylov, 28 pages. Version 2 differs from Version 1 only in improved presentation

Mader proved that for $k\geq 2$ and $n\geq 2k$, every $n$-vertex graph with no $(k+1)$-connected subgraphs has at most $(1+\frac{1}{\sqrt{2}})k(n-k)$ edges. He also conjectured that for $n$ large with respect to $k$, every such graph has at most $\frac{3}{2}\left(k - \frac{1}{3}\right)(n-k)$ edges. Yuster improved Mader's upper bound to $\frac{193}{120}k(n-k)$ for $n\geq\frac{9k}{4}$. In this note, we make the next step towards Mader's Conjecture: we improve Yuster's bound to $\frac{19}{12}k(n-k)$ for $n\geq\frac{5k}{2}$., 8 pages; a few typos have been fixed

Two graphs $G_{1} = (V_{1}, E_{1})$ and $G_{2} = (V_{2}, E_{2})$, each of order $n$, pack if there exists a bijection $f$ from $V_{1}$ onto $V_{2}$ such that $uv \in E_{1}$ implies $f(u)f(v) \notin E_{2}$. In 2014, ��ak proved that if $��(G_{1}), ��(G_{2}) \leq n-2$ and $|E_{1}| + |E_{2}| + \max \{ ��(G_{1}), ��(G_{2}) \} \leq 3n - 96n^{3/4} - 65$, then $G_{1}$ and $G_{2}$ pack. In the same paper, he conjectured that if $��(G_{1}), ��(G_{2}) \leq n-2$, then $|E_{1}| + |E_{2}| + \max \{ ��(G_{1}), ��(G_{2}) \} \leq 3n - 7$ is sufficient for $G_{1}$ and $G_{2}$ to pack. We prove that, up to an additive constant, ��ak's conjecture is correct. Namely, there is a constant $C$ such that if $��(G_1),��(G_2) \leq n-2$ and $|E_{1}| + |E_{2}| + \max \{ ��(G_{1}), ��(G_{2}) \} \leq 3n - C$, then $G_{1}$ and $G_{2}$ pack. In order to facilitate induction, we prove a stronger result on list packing., 21 pages, 11 figures

For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\le k+\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \le 6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$., Comment: 15 pages

An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-$k$-choosable bipartite graphs, showing that maximum average degree at most $2(k-1)$ is a sharp sufficient condition for $k$-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-$k$-colorable graphs and hypergraphs with any girth $g$., Comment: Slight adjustments, including simplifying the proofs of Lemmas 2.2 and 2.3 and the arguments for large girth

A graph $G$ is $k$-{\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. In a very recent paper, we gave a lower bound, $f_k(n) \geq F(k,n)$, that is sharp for every $n=1\,({\rm mod}\, k-1)$. It is also sharp for $k=4$ and every $n\geq 6$. In this note, we present a simple proof of the bound for $k=4$. It implies the case $k=4$ of the conjecture by Ore from 1967 that for every $k\geq 4$ and $n\geq k+2$, $f_k(n+k-1)=f(n)+\frac{k-1}{2}(k - \frac{2}{k-1})$. We also show that our result implies a simple short proof of the Gr\" otzsch Theorem that every triangle-free planar graph is 3-colorable.

A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with minimum color degree at least $k$ contains a rainbow matching of size at least $k$.

For a hypergraph G and a positive integer s, let \chi_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochv\'{i}l, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of \chi_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs., Comment: 9 pages

The Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G [] H of graphs. As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1 - o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We show that the above lower bound is asymptotically best possible. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let the (unique) prime factorization of G be given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k >= 2.log(log(chi(G))) + c', where c' is a constant. This improves the 2.log(chi(G))+3 bound of Chandran and Sivadasan. 2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >= c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if d >= 3.), 10 pages, 2 figures, major update: lower and upper bound proofs have been revised. The bounds are now asymptotically tight