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Jakub Przybyło

Publications and source records attributed to Jakub Przybyło.

At least 37 records · Page 2Linked to original sources

arXiv2022

On conflict-free proper colourings of graphs without small degree vertices

A proper vertex colouring of a graph $G$ is referred to as conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called $h$-conflict-free if there are at least $h$ such colours for each vertex of $G$. The least numbers of colours in such colourings of $G$ are denoted $χ_{\rm pcf}(G)$ and $χ_{\rm pcf}^h(G)$, respectively. It is known that $χ_{\rm pcf}^h(G)$ can be as large as $(h+1)(Δ+1)\approx Δ^2$ for graphs with maximum degree $Δ$ and $h$ very close to $Δ$. We provide several new upper bounds for these parameters for graphs with minimum degrees $δ$ large enough and $h$ detached from $δ$. In particular we show that $χ_{\rm pcf}^h(G)\leq (1+o(1))Δ$ if $δ\gg\lnΔ$ and $h\ll δ$, and that $χ_{\rm pcf}(G)\leq Δ+O(\ln Δ)$ for regular graphs. These specifically refer to the conjecture of Caro, Petruševski and Škrekovski that $χ_{\rm pcf}(G)\leq Δ+1$ for every connected graph $G$ of maximum degree $Δ\geq 3$, towards which they proved that $χ_{\rm pcf}(G)\leq \left\lfloor\frac{5Δ}{2}\right\rfloor$ if $Δ\geq 1$.

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arXiv2022

On triangle-free list assignments

We show that Bernshteyn's proof of the breakthrough result of Molloy that triangle-free graphs are choosable from lists of size $(1+o(1))Δ/\logΔ$ can be adapted to yield a stronger result. In particular one may prove that such list sizes are sufficient to colour any graph of maximum degree $Δ$ provided that vertices sharing a common colour in their lists do not induce a triangle in $G$, which encompasses all cases covered by Molloy's theorem. This was thus far known to be true for lists of size $(1000+o(1))Δ/\logΔ$, as implies a more general result due to Amini and Reed. We also prove that lists of length $2(r-2)Δ\log_2\log_2Δ/\log_2Δ$ are sufficient if one replaces the triangle by any $K_r$ with $r\geq 4$, pushing also slightly the multiplicative factor of $200r$ from Bernshteyn's result down to $2(r-2)$. All bounds presented are also valid within the more general setting of correspondence colourings.

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arXiv2022

A note on the conflict-free chromatic index

Let $G$ be a graph with maximum degree $Δ$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $χ'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $χ'_{CF}(G)\leq C_1\log_2Δ+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of graphs with $χ'_{CF}(G)\geq (1-o(1))\log_2Δ$. In this note we provide an explicit simple proof of the fact that $χ'_{CF}(G)\leq 3\log_2Δ+1$, which is a corollary of a stronger result: $χ'_{CF}(G)\leq 3\log_2χ(G)+1$. For this aim we prove a few auxiliary observations, implying in particular that $χ'_{CF}(G)\leq 4$ for bipartite graphs.

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arXiv2021

On a Problem of Steinhaus

Let $N$ be a positive integer. A sequence $X=(x_1,x_2,\ldots,x_N)$ of points in the unit interval $[0,1)$ is piercing if $\{x_1,x_2,\ldots,x_n\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. In 1958 Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most $74$ elements. This was later improved to the best possible value of $17$ by Warmus, and independently by Berlekamp and Graham. In this paper we study a more general variant of piercing sequences. Let $f(n)\geq n$ be an infinite nondecreasing sequence of positive integers. A sequence $X=(x_1,x_2,\ldots,x_{f(N)})$ is $f$-piercing if $\{x_1,x_2,\ldots,x_{f(n)}\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. A special case of $f(n)=n+d$, with $d$ a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each $d\geq 0$, the maximum length of any $(n+d)$-piercing sequence is finite. Expressing this maximum length as $s(d)+d$, they obtained an exponential upper bound on the function $s(d)$, which was later improved to $s(d)=O(d^3)$ by Graham and Levy. Recently, Konyagin proved that $2d\leqslant s(d)< 200d$ holds for all sufficiently big $d$. Using a different technique based on the Farey fractions and stick-breaking games, we prove here that the function $s(d)$ satisfies $\left\lfloor{}c_1d\right\rfloor{}\leqslant s(d)\leqslant c_2d+o(d)$, where $c_1=\frac{\ln 2}{1-\ln 2}\approx2.25$ and $c_2=\frac{1+\ln2}{1-\ln2}\approx5.52$. We also prove that there exists an infinite $f$-piercing sequence with $f(n)= γn+o(n)$ if and only if $γ\geq\frac{1}{\ln 2}\approx 1.44$.

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arXiv2021

On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted by $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ attributing distinct weighted degrees: $\tilde f(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ to all vertices $v\in V(G)$. It is straightforward that $s(G) \geq n/d$ for every $d$-regular graph $G$ on $n$ vertices with $d>1$. In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant $c$ such that $s(G) \leq n/d + c$ for all such graphs. Even though the conjecture has remained open in almost all relevant cases, it is more generally believed that there exists a universal constant $c$ such that $s(G) \leq n/δ+ c$ for every graph $G$ on $n$ vertices with minimum degree $δ\geq 1$ which does not contain an isolated edge. In this paper we confirm that the generalized Faudree-Lehel Conjecture holds for graphs with $δ\geq n^β$ where $β$ is any fixed constant larger than $0.8$. Furthermore, we confirm that the conjecture holds in general asymptotically. That is we prove that for any $\varepsilon\in(0,0.25)$ there exist absolute constants $c_1, c_2$ such that for all graphs $G$ on $n$ vertices with minimum degree %at least $δ\geq 1$ and without isolated edges, $s(G) \leq \frac{n}δ(1+\frac{c_1}{δ^{\varepsilon}})+c_2$, thus extending in various aspects and strengthening a recent result of Przybyło, who showed that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{ε/19}n})=\frac{n}{d}(1+o(1))$ for $d$-regular graphs with $d\in [\ln^{1+ε} n, n/\ln^εn]$, and improving an earlier general upper bound: $s(G)< 6\frac{n}δ+6$ of Kalkowski, Karoński and Pfender.

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arXiv2021

Short proof of the asymptotic confirmation of the Faudree-Lehel Conjecture

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ is unique amongst all $v\in V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) \leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) \geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d \geq n^{0.8+ε}$ for any fixed $ε>0$, with a small additive constant $c=28$ for $d$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $β\in(0,1/4)$ there is a constant $C$ such that for all $d$-regular graphs $G$, $s(G) \leq \frac{n}{d}(1+\frac{C}{d^β})+28$, extending and improving a recent result of Przybyło that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{ε/19}n})$ whenever $d\in [\ln^{1+ε} n, n/\ln^εn]$ and $d$ is large enough.

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arXiv2020

Conflict-free chromatic number vs conflict-free chromatic index

A vertex coloring of a given graph $G$ is conflict-free if the closed neighborhood of every vertex contains a unique color (i.e. a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict-free chromatic number of $G$, denoted $χ_{CF}(G)$. What is the maximum possible conflict-free chromatic number of a graph with a given maximum degree $Δ$? Trivially, $χ_{CF}(G)\leq χ(G)\leq Δ+1$, but it is far from optimal - due to results of Glebov, Szabó and Tardos, and of Bhyravarapu, Kalyanasundaram and Mathew, the answer in known to be $Θ\left(\ln^2Δ\right)$. We show that the answer to the same question in the class of line graphs is $Θ\left(\lnΔ\right)$ - that is, the extremal value of the conflict-free chromatic index among graphs with maximum degree $Δ$ is much smaller than the one for conflict-free chromatic number. The same result for $χ_{CF}(G)$ is also provided in the class of near regular graphs, i.e. graphs with minimum degree $δ\geq αΔ$.

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arXiv2020

The 1-2-3 Conjecture holds for graphs with large enough minimum degree

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a universal constant $K$, say $K=3$, such that one may always dispose of such pairs from any given connected graph with at least three vertices by blowing its selected edges into at most $K$ parallel edges? This question was first posed in 2004 by Karoński, Łuczak and Thomason, who equivalently asked if one may assign weights $1,2,3$ to the edges of every such graph so that adjacent vertices receive distinct weighted degrees - the sums of their incident weights. This basic problem is commonly referred to as the 1-2-3 Conjecture nowadays, and has been addressed in multiple papers. Thus far it is known that weights $1,2,3,4,5$ are sufficient [J. Combin. Theory Ser. B 100 (2010) 347-349]. We show that this conjecture holds if only the minimum degree $δ$ of a graph is large enough, i.e. when $δ= Ω(\logΔ)$, where $Δ$ denotes the maximum degree of the graph. The principle idea behind our probabilistic proof relies on associating random variables with a special and carefully designed distribution to most of the vertices of a given graph, and then choosing weights for major part of the edges depending on the values of these variables in a deterministic or random manner.

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arXiv2019

The 1-2-3 Conjecture almost holds for regular graphs

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set $\{1,2,3,4,5\}$. We show that for regular graphs it is sufficient to use weights $1$, $2$, $3$, $4$. Moreover, we prove the conjecture to hold for every $d$-regular graph with $d\geq 10^8$.

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arXiv2019

Asymptotic confirmation of the Faudree-Lehel Conjecture on irregularity strength for all but extreme degrees

The irregularity strength of a graph $G$, $s(G)$, is the least $k$ admitting a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ assuring distinct weighted degrees of all vertices, or equivalently the least possible maximal edge multiplicity in an irregular multigraph obtained of $G$ via multiplying some of its edges. The most well-known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists a constant $C$ such that $s(G)\leq \frac{n}{d}+C$ for each $d$-regular graph $G$ with $n$ vertices and $d\geq 2$ (while a straightforward counting argument yields $s(G)\geq \frac{n+d-1}{d}$). The best known results towards this imply that $s(G)\leq 6\lceil\frac{n}{d}\rceil$ for every $d$-regular graph $G$ with $n$ vertices and $d\geq 2$, while $s(G)\leq (4+o(1))\frac{n}{d}+4$ if $d\geq n^{0.5}\ln n$. We show that the conjecture of Faudree and Lehel holds asymptotically in the cases when $d$ is neither very small nor very close to $n$. We in particular prove that for large enough $n$ and $d\in [\ln^8n,\frac{n}{\ln^3 n}]$, $s(G)\leq \frac{n}{d}(1+\frac{8}{\ln n})$, and thereby we show that $s(G) = \frac{n}{d}(1+o(1))$ then. We moreover prove the latter to hold already when $d\in [\ln^{1+\varepsilon}n,\frac{n}{\ln^\varepsilon n}]$ where $\varepsilon$ is an arbitrary positive constant.

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arXiv2019

On the Standard (2,2)-Conjecture

The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every graph with minimum degree $δ\geq 10^6$ can be decomposed into two subgraphs requiring just weights $1$ and $2$ for the same goal. We thus prove the so-called Standard $(2,2)$-Conjecture for graphs with sufficiently large minimum degree. The result is in particular based on applications of the Lovász Local Lemma and theorems on degree-constrained subgraphs.

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arXiv2019

On inclusion chromatic index of a graph

Let $χ'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $δ\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident to any its neighbour. We provide an infinite family of examples of graphs $G$ with $χ'_\subset(G)\geq (1+\frac{1}{δ-1})Δ$, where $Δ$ is the maximum degree of $G$, and we conjecture that $χ'_\subset(G)\leq \lceil(1+\frac{1}{δ-1})Δ\rceil$ for every connected graph with $δ\geq 2$ which is not isomorphic to $C_5$. The equality here is attained e.g. for the family of complete bipartite graphs. Using a probabilistic argument we support this conjecture by proving that for any fixed $δ\ge2$, $χ'_\subset(G) \le (1+\frac{4}δ)Δ(1+o(1))$ (for $Δ\to\infty$), what implies that $χ'_\subset(G) \le (1+\frac{4}{δ-1})Δ$ for $Δ$ large enough.

math.CO↗

arXiv2018

Group twin coloring of graphs

For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $χ'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $χ'_g(G)\leq Δ(G)+3$ for all graphs without isolated edges, where $Δ(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $χ'_g(G)\leq 2(Δ(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$.

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arXiv2018

Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs

We investigate the \textit{group irregularity strength}, $s_g(G)$, of a graph, i.e. the least integer $k$ such that taking any Abelian group $\mathcal{G}$ of order $k$, there exists a function $f:E(G)\rightarrow \mathcal{G}$ so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on $s_g(G)$ for a general graph $G$ was exponential in $n-c$, where $n$ is the order of $G$ and $c$ denotes the number of its components. In this note we prove that $s_g(G)$ is linear in $n$, namely not greater than $2n$. In fact, we prove a stronger result, as we additionally forbid the identity element of a group to be an edge label or the sum of labels around a vertex. We consider also locally irregular labelings where we require only sums of adjacent vertices to be distinct. For the corresponding graph invariant we prove the general upper bound: $Δ(G)+{\rm col}(G)-1$ (where ${\rm col}(G)$ is the coloring number of $G$) in the case when we do not use the identity element as an edge label, and a slightly worse one if we additionally forbid it as the sum of labels around a vertex. In the both cases we also provide a sharp upper bound for trees and a constant upper bound for the family of planar graphs.

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arXiv2018

Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every $d$-regular graph, $d\geq 2$, can be decomposed into at most $2$ subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if $d\notin\{10,11,12,13,15,17\}$, and into at most $3$ such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most $24$ subgraphs fulfilling the 1-2-3 Conjecture, improving the previously best upper bound of $40$. Both results are partly based on applications of the Lovász Local Lemma.

math.CO↗

arXiv2018

On the total neighbour sum distinguishing index of graphs with bounded maximum average degree

A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c : V \cup E\to \{1,2,\ldots,k\}$ of colours to the edges and the vertices of $G$ such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing $k$-colouring, or tnsd $k$-colouring for short, is a proper total $k$-colouring such that $\sum_{e\ni u}c(e)+c(u)\neq \sum_{e\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by $χ''_Σ(G)$ the total neighbour sum distinguishing index of $G$, which is the least integer $k$ such that a tnsd edge $k$-colouring of $G$ exists. It has been conjectured that $χ''_Σ(G) \leq Δ(G) + 3$ for every graph $G$. In this paper we confirm this conjecture for any graph $G$ with ${\rm mad}(G)<\frac{14}{3}$ and $Δ(G) \geq 8$.

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arXiv2018

On the neighbour sum distinguishing index of planar graphs

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq Δ(G)$, while by Vizing's theorem, no more than $k=Δ(G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=Δ(G)+2$ enables enforcing additional strong feature of $c$, namely that it attributes distinct sums of incident colours to adjacent vertices in $G$ if only this graph has no isolated edges and is not isomorphic to $C_5$. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph $G$ of maximum degree at least $28$ which contains no isolated edges admits a proper edge colouring $c:E\to\{1,2,\ldots,Δ(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$.

cs.DM↗

arXiv2017

Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $Δ+CΔ^{\frac{1}{2}}(\logΔ)^4$, where $Δ$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.

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