arXiv subjects

Alexandr Kostochka

Publications and source records attributed to Alexandr Kostochka.

At least 55 records · Page 3Linked to original sources

arXiv2019

Fractional DP-Colorings of Sparse Graphs

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. In this paper we introduce and study the fractional DP-chromatic number $χ_{DP}^\ast(G)$. We characterize all connected graphs $G$ such that $χ_{DP}^\ast(G) \leqslant 2$: 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 $χ_\ell^\ast(G)$ of any graph $G$ equals its fractional chromatic number $χ^\ast(G)$. This equality does not extend to fractional DP-colorings. Moreover, we show that the difference $χ^\ast_{DP}(G) - χ^\ast(G)$ can be arbitrarily large, and, furthermore, $χ^\ast_{DP}(G) \geq d/(2 \ln d)$ 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.

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arXiv2019

Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization for large $n$ of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings.

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arXiv2019

Super-pancyclic hypergraphs and bipartite graphs

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.

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arXiv2019

Largest 2-regular subgraphs in 3-regular graphs

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.

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arXiv2019

Cubic graphs with small independence ratio

Let $i(r,g)$ denote the infimum of the ratio $\frac{α(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $α(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \ge 0.4375$, and Csóka improved it to $i(3,\infty) \ge 0.44533$ in 2016. Bollobás proved the upper bound $i(3,\infty) < \frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty) < 0.45537$ in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \le 0.454.$

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arXiv2019

Hypergraphs not containing a tight tree with a bounded trunk ~II: 3-trees with a trunk of size 2

A tight $r$-tree $T$ is an $r$-uniform hypergraph that has an edge-ordering $e_1, e_2, \dots, e_t$ such that for each $i\geq 2$, $e_i$ has a vertex $v_i$ that does not belong to any previous edge and $e_i-v_i$ is contained in $e_j$ for some $j<i$. Kalai conjectured in 1984 that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r}\binom{n}{r-1}$ edges contains every tight $r$-tree $T$ with $t$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree $T'$ of $T$ such that vertices in $V(T)\setminus V(T')$ are leaves in $T$. Kalai's Conjecture was proved in 1987 for tight $r$-trees that have a trunk of size one. In a previous paper we proved an asymptotic version of Kalai's Conjecture for all tight $r$-trees that have a trunk of bounded size. In this paper we continue that work to establish the exact form of Kalai's Conjecture for all tight $3$-trees with at least $20$ edges that have a trunk of size two.

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arXiv2019

Ordered and convex geometric trees with linear extremal function

The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ are linear in $n$, the latter posed by Braß-Károlyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in {\cal T}$ and $ex_{\rightarrow}(n,T) = Ω(n\log n)$ for $T \not\in {\cal T}$. We also describe the family of the convex geometric trees with linear Tur\' an number and show that for every convex geometric tree $F$ not in this family, $ex_{\cir}(n,F)= Ω(n\log \log n)$.

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arXiv2019

On 2-connected hypergraphs with no long cycles

We give an upper bound for the maximum number of edges in an $n$-vertex 2-connected $r$-uniform hypergraph with no Berge cycle of length $k$ or greater, where $n\geq k \geq 4r\geq 12$. For $n$ large with respect to $r$ and $k$, this bound is sharp and is significantly stronger than the bound without restrictions on connectivity. It turned out that it is simpler to prove the bound for the broader class of Sperner families where the size of each set is at most $r$. For such families, our bound is sharp for all $n\geq k\geq r\geq 3$.

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arXiv2018

Packing chromatic number of subdivisions of cubic graphs

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, $χ_p(G)$, 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 $χ_p(G)$ and of $χ_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $χ_p(D(G))\leq 5$ for any subcubic $G$, and later Bresar, Klavzar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $χ_p(G)$ is not bounded in the class of subcubic graphs $G$. In contrast, in this paper we show that $χ_p(D(G))$ is bounded in this class, and does not exceed $8$.

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arXiv2018

Extremal problems on ordered and convex geometric hypergraphs

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 Erdős-Ko-Rado Theorem, and generalizes earlier results of Capoyleas and Pach and Aronov-Dujmovič-Morin-Ooms-da Silveira.

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arXiv2018

Avoiding long Berge cycles II, exact bounds for all $n$

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $k\geq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $k\geq r+4$.

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arXiv2018

On $r$-uniform hypergraphs with circumference less than $r$

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 Győri, 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.

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arXiv2018

Avoiding long Berge cycles

Let $n\geq k\geq r+3$ and $\mathcal H$ be an $n$-vertex $r$-uniform hypergraph. We show that if $|\mathcal H|> \frac{n-1}{k-2}\binom{k-1}{r}$ then $\mathcal H$ contains a Berge cycle of length at least $k$. This bound is tight when $k-2$ divides $n-1$. We also show that the bound is attained only for connected $r$-uniform hypergraphs in which every block is the complete hypergraph $K^{(r)}_{k-1}$. We conjecture that our bound also holds in the case $k=r+2$, but the case of short cycles, $k\leq r+1$, is different.

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arXiv2018

A variation of a theorem by Pósa

A graph $G$ is $\ell$-hamiltonian if for any linear forest $F$ of $G$ with $\ell$ edges, $F$ can be extended to a hamiltonian cycle of $G$. We give a sharp upper bound for the maximum number of cliques of a fixed size in a non-$\ell$-hamiltonian graph. Furthermore, we prove stability for the bound: if a non-$\ell$-hamiltonian graph contains almost the maximum number of cliques, then it must be a subgraph of one of two examples.

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arXiv2018

Tight paths in convex geometric hypergraphs

In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz, Sutherland, Kupitz and Perles for convex geometric graphs, as well as the classical Erdős-Gallai Theorem for graphs. As a consequence, we obtain the first substantial improvement on the Turán problem for tight paths in uniform hypergraphs.

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arXiv2017

Hypergraphs not containing a tight tree with a bounded trunk

An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai [Kalai], generalizing the Erdős-Sós Conjecture for trees, asserts that if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform hypergraph containing no copy of $T$ then $G$ has at most $\frac{t-1}{r}\binom{n}{r-1}$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-T'$ has $r-1$ vertices in some edge of $T'$ and a vertex outside $T'$. For $r\ge 3$, the only nontrivial family of tight $r$-trees for which this conjecture has been proved is the family of $r$-trees with trunk size one in [FF] from 1987. Our main result is an asymptotic version of Kalai's conjecture for all tight trees $T$ of bounded trunk size. This follows from our upper bound on the size of a $T$-free $r$-uniform hypergraph $G$ in terms of the size of its shadow. We also give a short proof of Kalai's conjecture for tight $r$-trees with at most four edges. In particular, for $3$-uniform hypergraphs, our result on the tight path of length $4$ implies the intersection shadow theorem of Katona [Katona].

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arXiv2017

Sharp Dirac's Theorem for DP-Critical Graphs

Correspondence coloring, or DP-coloring, is a generalization of list coloring introduced recently by Dvořák and Postle. In this paper we establish a version of Dirac's theorem on the minimum number of edges in critical graphs in the framework of DP-colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz on classifying list-critical graphs that satisfy Dirac's bound with equality.

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arXiv2017

Many cliques in $H$-free subgraphs of random graphs

For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $χ(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|'\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $χ(H)-1$ partite subgraph of $G(n,p)$. The transition between these two behaviors occurs at $p=n^{-1/m_2(H)}$. When $m_2(H)< m_2(K_m)$ we show that the above cases still exist, however for $δ>0$ small at $p=n^{-1/m_2(H)+δ}$ one can typically still keep most of the copies of $K_m$ in an $H$-free subgraph of $G(n,p)$. Thus, the transition between the two behaviors in this case occurs at some $p$ significantly bigger than $n^{-1/m_2(H)}$. To show that the second case is not redundant we present a construction which may be of independent interest. For each $k \geq 4$ we construct a family of $k$ chromatic graphs $G(k,ε_i)$ where $m_2(G(k,ε_i))$ tends to $\frac{(k+1)(k-2)}{2(k-1)} (< m_2(K_{k-1}))$ as $i$ tends to infinity. This is tight for all values of $k$

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