Papers by Aleksandra Gorzkowska
arXiv (Cornell University), Dec 8, 2023
We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex... more We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial automorphism at least one edge changes its color. We consider these two notions together. We show that every graph without pendant edges has a majority distinguishing edge coloring with at most ⌈ √ ∆⌉ + 5 colors. Moreover, we show results for some classes of graphs and a general result for symmetric digraphs.
arXiv (Cornell University), Feb 25, 2024
A distinguishing index of a (di)graph is the minimum number of colours in an edge (or arc) colour... more A distinguishing index of a (di)graph is the minimum number of colours in an edge (or arc) colouring such that the identity is the only automorphism that preserves that colouring. We investigate the minimum and maximum value of the distinguishing index over all orientations of a given graph G. We present sharp results for these parameters in terms of the distinguishing index of G for trees, unbalanced bipartite graphs, traceable graphs and claw-free graphs. With this, we answer the question of Meslem and Sopena [8].
Theoretical Computer Science, Jul 1, 2017
Electronic Journal of Combinatorics, Aug 7, 2020
Ars Mathematica Contemporanea, May 20, 2016
arXiv (Cornell University), Oct 26, 2019
A graph G is said to be d-distinguishable if there is a vertex coloring of G with a set of d colo... more A graph G is said to be d-distinguishable if there is a vertex coloring of G with a set of d colors which breaks all of the automorphisms of G but the identity. We call the minimum d for which a graph G is d-distinguishiable the distinguishing number of G, denoted by D(G). When D(G) = 2, the minimum number of vertices in one of the color classes is called the cost of distinguishing of G and is shown by ρ(G). In this paper, we generalize this concept to edge-coloring by introducing the cost of edgedistinguishing of a graph G, denoted by ρ ′ (G). Then, we consider ρ ′ (K n) for n ≥ 6 by finding a procedure that gives the minimum number of edges of K n that should be colored differently to have a 2-distinguishing edge-coloring. Afterwards, we develop a machinery to state a sufficient condition for a coloring of the Cartesian product to break all non-trivial automorphisms. Using this sufficient condition, we determine when cost of distinguishing and edge-distinguishing of the Cartesian power of a path equals to one. We also show that this parameters are equal to one for any Cartesian product of finitely many paths of different lengths. Moreover, we do a similar work for the Cartesian powers of a cycle and also for the Cartesian products of finitely many cycles of different orders. Upper bounds for the cost of edge-distinguishing of hypercubes and the Cartesian powers of complete graphs are also presented.
Graphs with a unique maximum independent set up to automorphisms
Discrete Applied Mathematics
Discussiones Mathematicae Graph Theory, 2021
Ars Mathematica Contemporanea, 2021
A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a... more A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, γ pr ( G ) , of G is the minimum cardinality of a paired dominating set of G . A set of vertices whose removal from G produces a graph without isolated vertices is called a non-isolating set. The minimum cardinality of a non-isolating set of vertices whose removal decreases the paired domination number is the γ pr − -stability of G , denoted st γ pr − ( G ) . The paired domination stability of G is the minimum cardinality of a non-isolating set of vertices in G whose removal changes the paired domination number. We establish properties of paired domination stability in graphs. We prove that if G is a connected graph with γ pr ( G ) ≥ 4 , then st γ pr − ( G ) ≤ 2 Δ ( G ) where Δ ( G ) is the maximum degree in G , and we characterize the infinite fa...
arXiv: Combinatorics, 2019
A graph $G$ is said to be $d$-distinguishable if there is a vertex coloring of $G$ with a set of ... more A graph $G$ is said to be $d$-distinguishable if there is a vertex coloring of $G$ with a set of $d$ colors which breaks all of the automorphisms of $G$ but the identity. We call the minimum $d$ for which a graph $G$ is $d$-distinguishiable the distinguishing number of $G$, denoted by $D(G)$. When $D(G)=2$, the minimum number of vertices in one of the color classes is called the cost of distinguishing of $G$ and is shown by $\rho(G)$. In this paper, we generalize this concept to edge-coloring by introducing the cost of edge-distinguishing of a graph $G$, denoted by $\rho'(G)$. Then, we consider $\rho'(K_n )$ for $n\geq 6$ by finding a procedure that gives the minimum number of edges of $K_n$ that should be colored differently to have a $2$-distinguishing edge-coloring. Afterwards, we develop a machinery to state a sufficient condition for a coloring of the Cartesian product to break all non-trivial automorphisms. Using this sufficient condition, we determine when cost of dis...
The Electronic Journal of Combinatorics, 2020
The distinguishing index $D'(G)$ of a graph $G$ is the least number $k$ such that $G$ has an ... more The distinguishing index $D'(G)$ of a graph $G$ is the least number $k$ such that $G$ has an edge colouring with $k$ colours that is only preserved by the trivial automorphism. Pilśniak proved that a connected, claw-free graph has the distingushing index at most three. In this paper, we show that the distingushing index of a connected, claw-free graph with at least six vertices is bounded from above by two. We also consider more general graphs in this sense. Namely, we prove that if $G$ is a connected, $K_{1,s}$-free graph of order at least six, then $D'(G) \leq s-1$.
Ars Mathematica Contemporanea, 2016
The distinguishing index D (G) of a graph G is the least natural number d such that G has an edge... more The distinguishing index D (G) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k ≥ 2, the k-th Cartesian power of a connected graph G has distinguishing index equal 2, with the only exception D (K 2 2) = 3. We also prove that if G and H are connected graphs that satisfy the relation 2 ≤ |G| ≤ |H| ≤ 2 |G| 2 G − 1 − |G| + 2, then D (G2H) ≤ 2 unless G2H = K 2 2 .
Theoretical Computer Science, 2017
The distinguishing index D (G) of a graph G is the least number d such that G has an edge colouri... more The distinguishing index D (G) of a graph G is the least number d such that G has an edge colouring with d colours that is preserved only by the identity automorphism. The distinguishing index of the Cartesian product of graphs was investigated by the authors and Kalinowski. They considered colourings with two colours only and obtained results that do not determine the distinguishing index for all the possible cases. In this paper we investigate colourings with d colours and determine the exact value of the distinguishing index of the Cartesian product K 1,m 2K 1,n for almost all m and n. In particular, we supplement the result of [6] for the case when 2 2m+1 − m 2 + 1 < n ≤ 2 2m+1. We also observe the distinguishing index of the Cartesian product of two graphs in general does not have to depend on the size of the graphs and it can be arbitrarily small.
Graphs with a unique maximum independent set up to automorphisms
Discrete Applied Mathematics, Aug 1, 2022
Graphs and Combinatorics
A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a... more A set S of vertices in a graph G is a paired dominating set if every vertex of G is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, $$\gamma _{\mathrm{pr}}(G)$$ γ pr ( G ) , of G is the minimum cardinality of a paired dominating set of G. In this paper, we show that if T is a tree of order at least 2, then $$\gamma _{\mathrm{pr}}(T) \le 2\alpha (T) - \varphi (T)$$ γ pr ( T ) ≤ 2 α ( T ) - φ ( T ) where $$\alpha (T)$$ α ( T ) is the independence number and $$\varphi (T)$$ φ ( T ) is the $$P_3$$ P 3 -packing number. We present a tight upper bound on the paired domination number of a tree T in terms of its maximum degree $$\varDelta$$ Δ . For $$\varDelta \ge 1$$ Δ ≥ 1 , we show that if T is a tree of order n with maximum degree $$\varDelta$$ Δ , then $$\gamma _{\mathrm{pr}}(T) \le \left( \frac{5\varDelta -4}{8\varDelta -4} \right) n + \frac{1}{2}n_1(T) + \frac{1}{4}n_2(T) - \left...
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Papers by Aleksandra Gorzkowska