arXiv subjects

Sylwia Cichacz

Publications and source records attributed to Sylwia Cichacz.

At least 37 records · Page 2Linked to original sources

arXiv2022

$k$-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$

Vertex-fault-tolerance was introduced by Hayes~\cite{Hayes1976} in 1976, and since then it has been systematically studied in different aspects. In this paper we study $k$-vertex-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$, i.e., graphs in which removing any $k$ vertices leaves a graph that has $p$ disjoint complete graphs of order $c$ as a subgraph. The main contribution is to describe such graphs that have the smallest possible number of edges for $k=1$, $p \geq 1$, and $c \geq 3$. Moreover, we analyze some properties of such graphs for any value of $k$.

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arXiv2022

On b-acyclic chromatic number of a graph

Let $G$ be a graph. We introduce the acyclic b-chromatic number of $G$ as an analogue to the b-chromatic number of $G$. An acyclic coloring of a graph $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the induced subgraph on vertices of any two colors $i,j\in \{1,\dots,k\}$ induces a forest. On the set of all acyclic colorings of $G$ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number $A(G)$ of $G$ and the maximum cardinality of its minimal element is the acyclic b-chromatic number $A_b(G)$ of $G$. We present several properties of $A_b(G)$. In particular, we derive $A_b(G)$ for several known graph families, derive some bounds for $A_b(G)$, compare $A_b(G)$ with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.

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arXiv2021

On some graph-cordial Abelian groups

Hovey introduced $A$-cordial labelings as a generalization of cordial and harmonious labelings \cite{Hovey}. If $A$ is an Abelian group, then a labeling $f \colon V (G) \rightarrow A$ of the vertices of some graph $G$ induces an edge labeling on $G$; the edge $uv$ receives the label $f (u) + f (v)$. A graph $G$ is $A$-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. Patrias and Pechenik studied the larger class of finite abelian groups $A$ such that all path graphs are $A$-cordial. They posed a conjecture that all but finitely many paths graphs are $A$-cordial for any Abelian group $A$. In this paper we solve this conjecture. Moreover we show that all cycle graphs are $A$-cordial for any Abelian group $A$ of odd order.

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arXiv2020

Minimum k-critical bipartite graphs

We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, Δ_U, Δ_V)$, where $Δ_U$ and $Δ_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical bipartite if deleting at most $k=n-m$ vertices from $U$ creates $G'$ that has a complete matching, i.e., a matching of size $m$. We show that, if $m(n-m+1)/n$ is an integer, then a solution of the M$k$CBG-$(n,m)$ problem can be found among $(a,b)$-regular bipartite graphs of order $(n,m)$, with $a=m(n-m+1)/n$, and $b=n-m+1$. If $a=m-1$, then all $(a,b)$-regular bipartite graphs of order $(n,m)$ are $k$-critical bipartite. For $a<m-1$, it is not the case. We characterize the values of $n$, $m$, $a$, and $b$ that admit an $(a,b)$-regular bipartite graph of order $(n,m)$, with $b=n-m+1$, and give a simple construction that creates such a $k$-critical bipartite graph whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.

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arXiv2019

Group distance magic Cartesian product of two cycles

Let $G=(V,E)$ be a graph and $Γ$ an Abelian group both of order $n$. A $Γ$-distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow Γ$ for which there exists $μ\in Γ$ such that $% \sum_{x\in N(v)}\ell (x)=μ$ for all $v\in V$, where $N(v)$ is the neighborhood of $v$. Froncek %(\cite{ref_CicAus}) showed that the Cartesian product $C_m \square C_n$, $m, n\geq3$ is a $\mathbb{Z}_{mn}$-distance magic graph if and only if $mn$ is even. It is also known that if $mn$ is even then $C_m \square C_n$ has $\mathbb{Z}_α\times \mathcal{A}$-magic labeling for any $α\equiv 0 \pmod {{\rm lcm}(m,n)}$ and any Abelian group $\mathcal{A}$ of order $mn/α$. %\cite{ref_CicAus} However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution this problem by proving some necessary conditions. We further prove that for $n$ even the graph $C_{n}\square C_{n}$ has a $Γ$-distance magic labeling for any Abelian group $Γ$ of order $n^{2}$. Moreover we show that if $m\neq n$, then there does not exist a $(\mathbb{Z}_2)^{m+n}$-distance magic labeling of the Cartesian product $C_{2^m} \square C_{2^{n}}$. We also give necessary and sufficient condition for $C_{m} \square C_{n}$ with $\gcd(m,n)=1$ to be $Γ$-distance magic.

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arXiv2019

Realization of digraphs in Abelian groups and its consequences

Let $\overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $Γ$ be a finite Abelian group such that $|Γ|\geq 4|V(\overrightarrow{G})|$ or if $|V(\overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $\varepsilon>0$ then $|Γ|\geq (1+\varepsilon)|V(\overrightarrow{G})|$. We show that there exists an injective mapping $φ$ from $V(\overrightarrow{G})$ to the group $Γ$ such that $\sum_{x\in V(C)}φ(x) = 0$ for every connected component $C$ of $\overrightarrow{G}$, where $0$ is the identity element of $Γ$. Moreover we show some applications of this result to group distance magic labelings.

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arXiv2018

Note on the group edge irregularity strength of graphs

We investigate the \textit{edge group irregularity strength} ($es_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\mathcal{G}$ of order $s$, there exists a function $f:V(G)\rightarrow \mathcal{G}$ such that the sums of vertex labels at every edge are distinct. In this note we provide some upper bounds on $es_g(G)$ as well as for edge irregularity strength $es(G)$ and harmonious order $\rm{har}(G)$.

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arXiv2018

$Z_2\times Z_2$-cordial cycle-free hypergraphs

Hovey introduced $A$-cordial labelings as a generalization of cordial and harmonious labelings \cite{Hovey}. If $A$ is an Abelian group, then a labeling $f \colon V (G) \rightarrow A$ of the vertices of some graph $G$ induces an edge labeling on $G$, the edge $uv$ receives the label $f (u) + f (v)$. A graph $G$ is $A$-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. The problem of $A$-cordial labelings of graphs can be naturally extended for hypergraphs. It was shown that not every $2$-uniform hypertree (i.e., tree) admits a $Z_2\times Z_2$-cordial labeling \cite{Pechnik}. The situation changes if we consider $p$-uniform hypetrees for a bigger $p$. We prove that a $p$-uniform hypertree is $Z_2\times Z_2$-cordial for any $p>2$, and so is every path hypergraph in which all edges have size at least~3. The property is not valid universally in the class of hypergraphs of maximum degree~1, for which we provide a necessary and sufficient condition.

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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

A magic rectangle set on Abelian groups

A $Γ$-magic rectangle set $MRS_Γ(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $Γ$, each appearing once, with all row sums in every rectangle equal to a constant $ω\in Γ$ and all column sums in every rectangle equal to a constant $δ\in Γ$. In this paper we prove that for $\{a,b\}\neq\{2^α,2k+1\}$ where $α$ and $k$ are some natural numbers, a $Γ$-magic rectangle set MRS$_Γ(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Γ|$ is odd or $Γ$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $Γ$-magic rectangle MRS$_Γ(a, b)$=MRS$_Γ(a, b;1)$.

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arXiv2017

Note on group irregularity strength of disconnected graphs

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels at every vertex are distinct. So far it was not known if $s_g(G)$ is bounded for disconnected graphs. In the paper we we present some upper bound for all graphs. Moreover we give the exact values and bounds on $s_g(G)$ for disconnected graphs without a star as a component.

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arXiv2017

Zero sum partition into sets of the same order and its applications

We will say that an Abelian group $Γ$ of order $n$ has the $m$-\emph{zero-sum-partition property} ($m$-\textit{ZSP-property}) if $m$ divides $n$, $m\geq 2$ and there is a partition of $Γ$ into pairwise disjoint subsets $A_1, A_2,\ldots , A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$, where $g_0$ is the identity element of $Γ$. In this paper we study the $m$-ZSP property of $Γ$. We show that $Γ$ has $m$-ZSP if and only if $|Γ|$ is odd or $m\geq 3$ and $Γ$ has more than one involution. We will apply the results to the study of group distance magic graphs as well as to generalized Kotzig arrays.

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arXiv2016

Group Sum Chromatic Number of Graphs

We investigate the \textit{group sum chromatic number} ($\gchi(G)$) of graphs, i.e. the smallest value $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels properly colour the vertices. It is known that $\gchi(G)\in\{χ(G),χ(G)+1\}$ for any graph $G$ with no component of order less than $3$ and we characterize the graphs for which $\gchi(G)=χ(G)$.

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arXiv2015

Spectra of Graphs and Closed Distance Magic Labelings

Let $G=(V,E)$ be a graph of order $n$. A closed distance magic labeling of $G$ is a bijection $\ell \colon V(G)\rightarrow \{1,\ldots ,n\}$ for which there exists a positive integer $k$ such that $\sum_{x\in N[v]}\ell (x)=k$ for all $v\in V $, where $N[v]$ is the closed neighborhood of $v$. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic labelings and the spectra of graphs. These results are then applied to the strong product of graphs with complete graph or cycle and to the circulant graphs. We end with a number theoretic problem whose solution results in another family of closed distance magic graphs somewhat related to the strong product.

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arXiv2013

Note on group distance magic complete bipartite graphs

A $Γ$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $\ell$ from $V$ to an Abelian group $Γ$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}\ell(y)$ of every vertex $x \in V$ is equal to the same element $μ\in Γ$, called the \emph{magic constant}. A graph $G$ is called a \emph{group distance magic graph} if there exists a $Γ$-distance magic labeling for every Abelian group $Γ$ of order $|V(G)|$. In this paper we prove that some complete $k$-partite graphs are $\mathbb{Z}_p$-distance magic. Moreover we prove that $K_{m,n}$ is a group distance magic if and only if $n+m \not \equiv 2 \pmod 4$. We also show that if $n+m \equiv 2 \pmod 4$, then there does not exist a group $Γ$ of order $n+m$ such that there exists a $Γ$-distance labeling for $K_{m,n}$.

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arXiv2013

Group distance magic graphs $G\times C_n$

A $Γ$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $f$ from $V$ to an Abelian group $Γ$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element $μ\in Γ$, called the \emph{magic constant}. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that $°(v)\equiv c \imod {2^{p+2}}$ for some constant $c$ for any $v\in V(G)$, then there exists a $Γ$-distance magic labeling for any Abelian group $Γ$ of order $4n$ for the direct product $G\times C_4$. Moreover if $c$ is even then there exists a $Γ$-distance magic labeling for any Abelian group $Γ$ of order $8n$ for the direct product $G\times C_8$.

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arXiv2012

Distance magic labeling and two products of graphs

Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$ is the neighborhood of $v$. We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.

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