arXiv subjects

Aleksandra Gorzkowska

Publications and source records attributed to Aleksandra Gorzkowska.

6 recordsLinked to original sources

arXiv2026

Arc-distinguishing of orientations of graphs

The distinguishing index $D'(G)$ of a graph $G$ is the minimum number of colours in an edge colouring preserved only by the identity automorphism. We study how orienting the edges affects this parameter, relating the minimum and maximum distinguishing indices over all orientations of $G$ to $D'(G)$. We establish bounds and exact relations for bipartite graphs and trees, and study rigid orientations of traceable and claw-free graphs. Our results answer a question of Meslem and Sopena on unbalanced complete bipartite graphs.

math.CO↗

arXiv2026

Distinguishing adjacent vertices by ordering edges

The 1-2-3 Conjecture states that for every graph without isolated edges, there exists an edge-weighting from $\{1,2,3\}$ such that adjacent vertices receive distinct sums of weights on their incident edges. In the sequence variant, adjacent vertices are to be distinguished by the sequences of weights on their incident edges. In this paper, we investigate whether, for every graph without isolated edges and for a fixed proper edge colouring (where colours are interpreted as weights), there exists a global total order of the edges such that the resulting sequences of incident weights distinguish adjacent vertices. For connected graphs, we prove that such an order exists whenever there exist two adjacent vertices that have distinct sets of incident weights. This yields a positive answer for every proper edge colouring of a connected non-regular graph or a connected graph of class two. Moreover, a probabilistic argument gives the same conclusion for connected regular graphs of degree at least six.

math.CO↗

arXiv2026

Locally Irregular Total Colorings of Graphs

A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of $\binom{V_0 \cup V_1}{2}$ $(E\cap(V_0 \cup V_1)=\emptyset)$. A simple graph is a total graph in which all vertices are full. We say that a total graph $G$ is locally irregular if every two adjacent vertices have different total degrees, where by the total degree of a vertex $v$ in $G$ we mean the number of edges in $G$ that contain $v$ plus 1 if $v$ is full, or plus 0 if $v$ is empty. A total coloring of a graph $G$ whose colors induce locally irregular total subgraphs is called locally irregular total coloring, and the minimum number of colors required in such a coloring of $G$ is denoted by ${\rm tlir}(G)$. In 2015, Baudon, Bensmail, Przybyło, and Woźniak conjectured that ${\rm tlir}(G)\leq 2$ for every graph $G$. In this paper, we prove this conjecture for cacti, subcubic graphs, and split graphs. We also provide a general upper bound for ${\rm tlir}(G)$ depending on the chromatic number of $G$, and a constant upper bound if $G$ is planar or outerplanar. In our proofs, we utilize special decompositions of graphs and the connection between acyclic vertex coloring and locally irregular total coloring.

math.CO↗

arXiv2024

A note on sequences variant of irregularity strength for hypercubes

Let $f: E \mapsto \{1,2,\dots,k\}$ be an edge coloring of the $n$ - dimensional hypercube $H_n$. By the palette at a vertex $v$ we mean the sequence $\left(f(e_1(v)), f(e_1(v)),\dots, f(e_n(v))\right)$, where $e_i(v)$ is the $i$ - dimensional edge incident to $v$. In the paper, we show that two colors are enough to distinguish all vertices of the $n$ - dimensional hypercube $H_n$ ($n \geq 2$) by their palettes. We also show that if $f$ is a proper edge coloring of the hypercube $H_n$ ($n\geq 5$), then $n$ colors suffice to distinguish all vertices by their palettes.

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arXiv2023

Majority distinguishing edge coloring

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 $\lceil\sqrtΔ\rceil+5$ colors. Moreover, we show results for some classes of graphs and a~general result for symmetric digraphs.

math.CO↗

arXiv2019

The Cost of Edge-distinguishing of the Cartesian Product of Connected Graphs

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 edge-distinguishing of a graph $G$, denoted by $ρ'(G)$. Then, we consider $ρ'(K_n )$ 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 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.

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