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Andrzej Czygrinow

Publications and source records attributed to Andrzej Czygrinow.

At least 19 recordsLinked to original sources
arXiv2026

Rainbow cycles in triangle-free graphs

Let $G = (V,E)$ be an edge-colored graph, and let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$. We show that for a sufficiently large $n$ if $G$ is an edge-colored triangle-free graph of order $n$ that satisfies $δ^c(G)\geq (n+7)/5$, then $G$ contains a rainbow cycle of length four, which improves a bound of Ding et al. and is best possible. In addition, we show that given $k$, there is $n_0$ such that for $n\geq n_0$, if $G$ is an edge-colored triangle-free graph with $δ^c(G)> n/5+3$, then $G$ contains a rainbow cycle of length $4k$.

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arXiv2024

Tight minimum colored degree condition for rainbow connectivity

Let $G = (V, E)$ be a graph on $n$ vertices, and let $c: E \to P$, where $P$ is a set of colors. Let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $δ^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. In this paper, we show that the same bound for $δ^c(G)$ implies that any two vertices are connected by a rainbow path.

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arXiv2024

Rainbow Cliques in Edge-Colored Graphs

Let $G = (V,E)$ be an $n$-vertex graph and let $c: E \to \mathbb{N}$ be a coloring of its edges. Let $d^c(v)$ be the number of distinct colors on the edges at $v \in V$ and let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$. H. Li proved that $δ^c(G) > n/2$ guarantees a rainbow triangle in $G$. We give extensions of Li's result to cliques $K_r$ for $r \ge 4$.

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arXiv2022

Distributed distance domination in graphs with no $K_{2,t}$-minor

We prove that a simple distributed algorithm finds a constant approximation of an optimal distance-$k$ dominating set in graphs with no $K_{2,t}$-minor. The algorithm runs in a constant number of rounds. We further show how this procedure can be used to give a distributed algorithm which given $ε>0$ and $k,t\in \mathbb{Z}^+$ finds in a graph $G=(V,E)$ with no $K_{2,t}$-minor a distance-$k$ dominating set of size at most $(1+ε)$ of the optimum. The algorithm runs in $O(\log^*{|V|})$ rounds in the Local model. In particular, both algorithms work in outerplanar graphs.

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arXiv2021

On Odd Rainbow Cycles in Edge-Colored Graphs

Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis ensures a rainbow $\ell$-cycle $C_{\ell}$ whenever $n \ge 432 \ell$. This result is sharp for all odd integers $\ell \geq 3$, and extends earlier work of the authors for when $\ell$ is even.

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arXiv2019

On Even Rainbow or Nontriangular Directed Cycles

Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We establish a corresponding result for fixed even rainbow $\ell$-cycles $C_{\ell}$: if every vertex $v \in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n \geq n_0(\ell)$ is sufficiently large, then $G$ admits an even rainbow $\ell$-cycle $C_{\ell}$. This result is best possible whenever $\ell \not\equiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer $\ell \geq 4$, every large $n$-vertex oriented graph $\vec{G} = (V, \vec{E})$ with minimum outdegree at least $(n+1)/3$ admits a (consistently) directed $\ell$-cycle $\vec{C}_{\ell}$. Our latter result relates to one of Kelly, Kühn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree. Our proofs are based on the stability method.

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arXiv2018

Even cycles in dense graphs

We will show that for $α>0$ there is $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices such that $αn< δ(G)< (n-1)/2$, then for every $n_1+n_2+\cdots +n_l= δ(G)$, $G$ contains a disjoint union of $C_{2n_1},C_{2n_2}, \dots, C_{2n_l}$ unless $G$ has a very specific structure.

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arXiv2018

Optimal pebbling number of graphs with given minimum degree

Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number $π^*(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\frac{4n}{δ+1}$, where $δ$ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If $\operatorname{diam}(G)\geq 3$ then we further improve the bound to $π^*(G)\leq\frac{3.75n}{δ+1}$. On the other hand, we show that for arbitrary large diameter and any $ε>0$ there are infinitely many graphs whose optimal pebbling number is bigger than $\left(\frac{8}{3}-ε\right)\frac{n}{(δ+1)}$.

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arXiv2017

Balanced Spanning Caterpillars

A $p$-caterpillar is a caterpillar such that every non-leaf vertex is adjacent to exactly $p$ leaves. We give a tight minimum degree condition for a graph to have a spanning $p$-caterpillar.

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arXiv2016

Tiling directed graphs with tournaments

The Hajnal--Szemerédi theorem states that for any integer $r \ge 1$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $δ(G) \ge (1 - 1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any integer $r \ge 4$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1 - 1/r)n - 1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size $r$ each of which contain every tournament on $r$ vertices. A related Turán-type result is also proven.

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arXiv2013

Tiling in bipartite graphs with asymmetric minimum degrees

The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladký and Schacht, and Czygrinow and DeBiasio determined the optimal minimum degree condition for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t} for fixed positive integers s<t. For a balanced bipartite graph G[U,V], let δ_U be the minimum degree over all vertices in U and δ_V be the minimum degree over all vertices in V. We consider the problem of determining the optimal value of δ_U+δ_V which guarantees that G can be tiled with K_{s,s}. We show that the optimal value depends on D:=|δ_V-δ_U|. When D is small, we show that δ_U+δ_V\geq n+3s-5 is best possible. As D becomes larger, we show that δ_U+δ_V can be made smaller, but no smaller than n+2s-2s^{1/2}. However, when D=n-C for some constant C, we show that there exist graphs with δ_U+δ_V\geq n+s^{s^{1/3}} which cannot be tiled with K_{s,s}.

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arXiv2013

On directed versions of the Corrádi-Hajnal Corollary

For $k \in \mathbb N$, Corrádi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $δ(G) \ge 2k$ has a $C_3$-factor, i.e., a partitioning of the vertex set so that each part induces the 3-cycle $C_3$. Wang proved that every directed graph $\overrightarrow G$ on $3k$ vertices with minimum total degree $δ_t(\overrightarrow G):=\min_{v\in V}(deg^-(v)+deg^+(v)) \ge 3(3k-1)/2$ has a $\overrightarrow C_3$-factor, where $\overrightarrow C_3$ is the directed 3-cycle. The degree bound in Wang's result is tight. However, our main result implies that for all integers $a \ge 1$ and $b \ge 0$ with $a+b=k$, every directed graph $\overrightarrow G$ on $3k$ vertices with minimum total degree $δ_t(\overrightarrow G)\ge 4k-1$ has a factor consisting of $a$ copies of $\overrightarrow T_3$ and $b$ copies of $\overrightarrow C_3$, where $\overrightarrow T_3$ is the transitive tournament on three vertices. In particular, using $b=0$, there is a $\overrightarrow T_3$-factor of $\overrightarrow G $, and using $a=1$, it is possible to obtain a $\overrightarrow C_3$-factor of $\overrightarrow G$ by reversing just one edge of $\overrightarrow G$. All these results are phrased and proved more generally in terms of undirected multigraphs. We conjecture that every directed graph $\overrightarrow G$ on $3k$ vertices with minimum semidegree $δ_0(\overrightarrow G):=\min_{v\in V}\min(deg^-(v),deg^+(v)) \ge 2k$ has a $\overrightarrow C_3$-factor, and prove that this is asymptotically correct.

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arXiv2013

An extension of the Hajnal-Szemeredi theorem to directed graphs

Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

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arXiv2012

Tiling 3-uniform hypergraphs with K_4^3-2e

Let K_4^3-2e denote the hypergraph consisting of two triples on four points. For an integer n, let t(n, K_4^3-2e) denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pair-degree δ_2(G) \geq d contains \floor{n/4} vertex-disjoint copies of K_4^3-2e. Kühn and Osthus proved that t(n, K_4^3-2e) = (1 + o(1))n/4 holds for large integers n. Here, we prove the exact counterpart, that for all sufficiently large integers n divisible by 4, t(n, K_4^3-2e) = n/4 when n/4 is odd, and t(n, K_4^3-2e) = n/4+1 when n/4 is even. A main ingredient in our proof is the recent `absorption technique' of Rödl, Ruciński and Szemerédi.

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arXiv2011

A note on bipartite graph tiling

Bipartite graph tiling was studied by Zhao who gave the best possible minimum degree conditions for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s}. Let s 2s+1. We give the best possible minimum degree condition in this case.

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arXiv2004

Girth, Pebbling, and Grid Thresholds

In this note we answer a question of Hurlbert about pebbling in graphs of high girth. Specifically we show that for every g there is a Class 0 graph of girth at least g. The proof uses the so-called Erdos construction and employs a recent result proved by Czygrinow, Hurlbert, Kierstead and Trotter. We also use the Czygrinow et al. result to prove that Graham's pebbling product conjecture holds for dense graphs. Finally, we consider a generalization of Graham's conjecture to thresholds of graph sequences and find reasonably tight bounds on the pebbling threshold of the sequence of d-dimensional grids, verifying an important instance the generalization.

math.CO↗
arXiv2004

Pebbling in Dense Graphs

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertices having minimum degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336 vertices in each part having minimum degree at least floor(m/2)+1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when d is proportional to n.

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