Mariusz Grech, Andrzej Kisielewicz
The distinguishing number $D(Γ)$ of a graph $Γ$ is the least size of a partition of the vertices of $Γ$ such that no non-trivial automorphism of $Γ$ preserves this partition. We show that if the automorphism group of a graph $Γ$ is simple, than $D(Γ)=2$. This is obtained by establishing the distinguishing number for all possible actions of simple groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $Γ$ be a group acting on a set $X$. The distinguishing number for this action of $Γ$ on $X$, denoted by $D_Γ(X)$, is the smallest natural number $k$ such that the elements of $X$ can be labeled with $k$ labels so that any label-preserving element of $Γ$ fixes all $x \in X$. In particular, if the action is faithful, then the only element of $Γ$ preserving labels is the identity. In this paper, we obtain an upper bound on the distinguishing number of a set knowing the distinguishing number of a set under the action of a subgroup. By the concept of motion, we obtain an upper bound for the dis…
The distinguishing number of complete bipartite and crown graphs
研究完全二分图和皇冠图的区分数，确定其自同构群的区分数范围及相关子群的性质。
The distinguishing number of a structure is the smallest size of a partition of its elements so that only the trivial automorphism of the structure preserves each cell of the partition. We show that for any countable subset of the positive real numbers, the corresponding countable homogeneous Urysohn metric space, when it exists, has distinguishing number 2 or the distinguishing number is infinite. While it is known that a sufficiently large finite primitive structure has distinguishing number 2, unless its automorphism group is the full symmetric group or alternating group, the infinite case …
The distinguishing number of a graph was introduced by Albertson and Collins as a measure of the amount of symmetry contained in the graph. Tymoczko extended this definition to faithful group actions on sets; taking the set to be the vertex set of a graph and the group to be the automorphism group of the graph allows one to recover the previous definition. Since then, several authors have studied properties of the distinguishing number as well as extensions of the notion. In this paper, we first answer a few open questions regarding the distinguishing number. Next we turn to generalizations re…
A \textit{distinguishing partition} of a group $X$ with automorphism group ${aut}(X)$ is a partition of $X$ that is fixed by no nontrivial element of ${aut}(X)$. In the event that $X$ is a complete multipartite graph with its automorphism group, the existence of a distinguishing partition is equivalent to the existence of an asymmetric hypergraph with prescribed edge sizes. An asymptotic result is proven on the existence of a distinguishing partition when $X$ is a complete multipartite graph with $m_1$ parts of size $n_1$ and $m_2$ parts of size $n_2$ for small $n_1$, $m_2$ and large $m_1$, $n…
A graph G is distinguished if its vertices are labelled by a map ϕ: V(G) \longrightarrow {1,2,...,k} so that no graph automorphism preserves ϕ. The distinguishing number of G is the minimum number k necessary for ϕto distinguish the graph. It is one measure of the complexity of the graph. We extend these definitions to an arbitrary group action of G on a set X. A labelling ϕ: X \longrightarrow {1,2,...,k} is distinguishing if no nontrivial element of G preserves ϕexcept those in the stabilizer of X. The distinguishing number of the group action on X is the minimum k needed for ϕto distinguish …
The distinguishing number $D(G,X)$ of an action of a group $G$ on a set $X$ is the least size of a partition of $X$ such that no element of $G$ acting nontrivially on $X$ preserves this partition. In this paper we describe the distinguishing numbers for all actions of the symmetric group $S_n$, for any $n\geq 3$. This allows us to describe the distinguishing numbers for all graphs whose automorphism group is isomorphic with a symmetric group. Our description solves a few open problems posed by various authors in earlier papers on this topic.
Let $G$ be a permutation group acting on a set $V$. A partition $π$ of $V$ is distinguishing if the only element of $G$ that fixes each cell of $π$ is the identity. The distinguishing number of $G$ is the minimum number of cells in a distinguishing partition. We prove that if $G$ is a primitive permutation group and $|V|\ge336$, its distinguishing number is two.
Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we show that if G is nilpotent of class c or supersolvable of length c then G always acts with distinguishing number at most c+1. We obtain that all metacyclic groups act with distinguishing number at most 3; these include all groups of squarefree order. We also prove that the distinguishing number of the action of the general linear g…
Given a group $Γ$ acting on a set $X$, a $k$-coloring $ϕ:X\to\{1,\dots,k\}$ of $X$ is distinguishing with respect to $Γ$ if the only $γ\in Γ$ that fixes $ϕ$ is the identity action. The distinguishing number of the action $Γ$, denoted $D_Γ(X)$, is then the smallest positive integer $k$ such that there is a distinguishing $k$-coloring of $X$ with respect to $Γ$. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph $G$ upon its vertex set, which is refe…
An adjacent vertex distinguishing total $k$-coloring $f$ of a graph $G$ is a proper total $k$-coloring of $G$ such that no pair of adjacent vertices has the same color sets, where the color set at a vertex $v$, $C^G_f(v)$, is $\{f(v)\} \cup \{f(vu)|u \in V (G), vu \in E(G)\}$. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph $G$ has adjacent vertex distinguishing total $(Δ(G)+3)$-coloring. In this paper we confirm the conjecture for many coronas, in particular for generalized, simple and $l$-coronas of graphs, not relating the results to particular graph classes.
Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product of G and H in terms of the number of distinguishing colorings of X with respect to G and the disting…
We show that for all simple graphs G other than the cycles C_3,C_4,C_5, and the claw K_1,3 there exists a K > 0 such that whenever k > K the k-th iterate of the line graph can be distinguished by at most two colors. Additionally we determine, for trees, when the distinguishing number of the line graph of T is greater than the distinguishing number of T.
Upper bounds for the list-distinguishing chromatic number
本文针对简单有限连通图，证明了Brooks定理在列表可区分色数上的类比，并利用着色数和列表色数给出两个上界，同时确定了书图等多种图族的列表可区分色数。
An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree $Δ$ and with no isolated edges has an \avd-coloring with at most $Δ+300$ colors, provided that $Δ>10^{20}$.
The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $Γ(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-Ω(\log n)}$ we have, $χ_D(Γ)\leχ(Γ) + 1$.
A non-zero component graph $G(\mathbb{V})$ associated to a finite vector space $\mathbb{V}$ is a graph whose vertices are non-zero vectors of $\mathbb{V}$ and two vertices are adjacent, if their corresponding vectors have at least one non-zero component common in their linear combination of basis vectors. In this paper, we extend the study of properties of automorphisms of non-zero component graphs. We prove that every permutation of basis vectors can be extended to an automorphism of $G(\mathbb{V})$. We prove that the symmetric group of basis vectors of $\mathbb{V}$ is isomorphic to the autom…
A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k parts such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G,V) is k-distinguishable is its distinguishing number. In particular, a graph X is k-distinguishable if its automorphism group Aut(X) has distinguishing number at most k in its action on the vertices of X. Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficien…
Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In fact, we prove that chi D(G) \leq chi L(G). Moreover, we determine some types of graphs whose locating and distinguishing chromatic numbers are equal. Specially, we characteristic all graph G with the property that chi D(G)= chi L(G) = 3.
The distinguishing index of a simple graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. It was conjectured by Pilśniak (2015) that for any 2-connected graph $D'(G) \leq \lceil \sqrt{Δ(G)}\rceil +1$. We prove a more general result for the distinguishing index of graphs with minimum degree at least two from which the conjecture follows. Also we present graphs $G$ for which $D'(G)\leq \lceil \sqrt{Δ}\rceil$.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A graphoidal cover of $G$ is a collection $ψ$ of (not necessarily open) paths in $G$ such that every path in $ψ$ has at least two vertices, every vertex of $G$ is an internal vertex of at most one path in $ψ$ and every edge of $G$ is in exactly one path in $ψ$. Let $Ω(G,ψ)$ denote the intersection graph of $ψ$. A graph $H$ is called a graphoidal graph, if there exists a graph $G$ and a g…
Suppose that $χ_λ$ and $χ_μ$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial in $n$, constructs $π\in S_n$ such that $χ_λ(π)$ is provably different from $χ_μ(π)$. In fact, we show a little more. Suppose $f=χ_λ$ for some irreducible character $χ_λ$ of $S_n$, but we do not know $λ$, and we are given only oracle access to $f$. We give an algorithm that determines $λ$, using a number of queries to $f$ that is polynomial in $n$. Each query can be computed in time polynomial in $n$ by someone who knows $λ$.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The lexicographic product of two graphs $G$ and $H$, $G[H]$ can be obtained from $G$ by substituting a copy $H_u$ of $H$ for every vertex $u$ of $G$ and then joining all vertices of $H_u$ with all vertices of $H_v$ if $uv\in E(G)$. In this paper we obtain some sharp bounds for the distinguishing number and the distinguishing index of lexicographic product of two graphs. As consequences, …
Automorphism groups and Distinguishing Colorings of Central and Middle Graphs
本文证明当原图阶数至少为4时，其自同构群与中心图、中间图的自同构群同构，并利用该结果给出中心图和中间图的区分数与区分指数的新上界，同时通过幂等交换拉丁方和Galvin定理研究中心图的全区分色数。
The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The {\em distinguishing number} of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph $Γ$ is said to have {\em connectivity 1} if there exists a vertex $α\in VΓ$ such that $Γ\setminus \{α\}$ is not connected. For $α\in V$, an orbit of the point stabilizer $G_α$ is called a {\em suborbit} of $G$. We prove that every connected…
For a graph $G$ and a positive integer $k$, a vertex labelling $f:V(G)\to\{1,2\ldots,k\}$ is said to be $k$-distinguishing if no non-trivial automorphism of $G$ preserves the sets $f^{-1}(i)$ for each $i\in\{1,\ldots,k\}$. The distinguishing chromatic number of a graph $G$, denoted $χ_D(G)$, is defined as the minimum $k$ such that there is a $k$-distinguishing labelling of $V(G)$ which is also a proper coloring of the vertices of $G$. In this paper, we prove the following theorem: Given $k\in\mathbb{N}$, there exists an infinite sequence of vertex-transitive graphs $G_{i}=(V_i,E_i)$ such that …
Let $δ$ and $Δ$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. Motivated by a conjecture by Pilśniak (2017) that implies that for any $2$-connected graph $D'(G) \leq \lceil \sqrt{Δ(G)}\rceil +1$, we prove that for any graph $G$ with $δ\geq 2$, $D'(G) \leq \lceil \sqrt[δ]{Δ}\rceil +1$. Also, we show that the distinguishing index of $k$-regular graphs is at most $2$, for any $k\ge…
Given a proper total $k$-coloring $c:V(G)\cup E(G)\to\{1,2,\ldots,k\}$ of a graph $G$, we define the value of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose values form a proper coloring is the neighbor sum distinguishing total chromatic number of $G$, $χ"_Σ(G)$. Pilśniak and Woźniak (2013) conjectured that $χ"_Σ(G)\leq Δ(G)+3$ for any simple graph with maximum degree $Δ(G)$. In this paper, we prove this bound to be asymptotically correct by showing that $χ"_Σ(G)\leq Δ(G)(1+o(1))$. The main idea of our argument…
The distinguishing index $D'(Γ)$ of a graph $Γ$ is the least number $k$ such that $Γ$ has an edge-coloring with $k$ colors preserved only by the trivial automorphism. In this paper we prove that if the automorphism group of a finite graph $Γ$ is simple, then its distinguishing index $D'(Γ)=2$.
An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that edge weighting problem has a close relationship with special factors of graphs. In this paper, we obtain several results on the existence of factors with the pre-specified degrees, which generalizes earlier results in [2, 3]. Using these results, we investigate edge-weighting problem. In particular, we prove that every 4-colorable graph admits a vertex-coloring 4-edge-weight…
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. For any $n \in \mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$. The $m^{th}$ power of $G$, is a graph with same set of vertices of $G$ and an edge between two vertices if and only if there is a path of length at most $m$ between them. The fractional power of $G$, denoted by $G^{\frac{…
The distinguishing number $\operatorname D(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has a labeling with $d$ labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs $G$ with infinite motion and growth $o \left(\frac{n^2}{\log_2 n} \right)$ is either $1$ or $2$, which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs $G$ of arbitrary cardinality are $2$-dist…
A colouring of a graph G is called distinguishing if its stabiliser in Aut G is trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabiliser of such a colouring is almost surely nowhere dense in Aut G and a null set with respect to the Haar measure on the automorphism group. We also investigate random 2-colourings in several classes of locally finite graphs where the existence of a distinguishing 2…
Fixing Numbers of Graphs and Groups
本文定义了图与群的固定数及固定集，将其视为图区分数的变体，并刻画了有限阿贝尔群的固定集，同时研究了对称群的固定集。
Distinguishing graphs with simple spectrum by homomorphism counts
本研究探讨了通过同态计数区分具有简单谱的图，证明了此类图类需满足无界树宽等条件，并构造了相关图对，扩展了三次同态计数的应用范围。
In the setting of a group $Γ$ acting faithfully on a set $X$, a $k$-coloring $c: X\rightarrow \{1, 2, ..., k\}$ is called $Γ$-distinguishing if the only element of $Γ$ that fixes $c$ is the identity element. The distinguishing number $D_Γ(X)$ is the minimum value of $k$ such that a $Γ$-distinguishing $k$-coloring of $X$ exists. Now, fixing $k= D_Γ(X)$, a subset $W\subset X$ with trivial pointwise stabilizer satisfies the precoloring extension property $P(W)$ if every precoloring $c: X-W\rightarrow \{1, ..., k\}$ can be extended to a $Γ$-distinguishing $k$-coloring of $X$. The distinguishing ex…
Distinguishing chromatic number of middle and subdivision graphs
本文研究了中间图和细分图的区分色数，通过应用已有结果确定中间图的区分色数，并证明细分图的区分数与总区分数相关，给出了细分图的区分色数上界。
A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the distinguishing threshold of $G$, denoted by $θ(G)$, is the minimum number $k$ such that every $k$-coloring of $G$ is distinguishing. As an alternative definition, $θ(G)$ is one more than the maximum number of cycles in the cycle decomposition of automorphisms of $G$. In this paper, we characterize $θ(G)$ when $G$ is disconnected. Afterwards, we prove that, alt…
The List-distinguishing chromatic number of graphs containing only small complete bigraphs
本文受Catlin组合技术启发，为仅含小完全二部图导出子图的连通图类建立了可区别色数与列表可区别色数的改进上界，并应用Menger定理展示了结果在基于Paley图、Dihedral群Cayley图等构造的图中的应用。
A New Game Invariant of Graphs: the Game Distinguishing Number
本文提出图的博弈区分数，定义温和者与顽皮者的对抗博弈，给出无穷性条件、对合图二次上界，并计算超立方体、偶圈及非素数阶奇圈的精确值，素数阶奇圈上界为3。
Adjacent vertex distinguishing total chromatic number of graph products
本文针对笛卡尔乘积、字典序乘积等多类图乘积，验证了邻点可区别全染色猜想（AVD-TCC）的正确性，拓展了该图论猜想的适用范围。
Consider a simple graph $G=(V,E)$ of maximum degree $Δ$ and its proper total colouring $c$ with the elements of the set $\{1,2,\ldots,k\}$. The colouring $c$ is said to be \emph{neighbour sum distinguishing} if for every pair of adjacent vertices $u$, $v$, we have $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ for which it exists is denoted by $χ"_{\sum}(G)$, hence $χ"_{\sum}(G) \geq Δ+1$. On the other hand, it has been daringly conjectured that just one more label than presumed in the famous Total Colouring Conjecture suffices to construct such total colouring $c$,…
The least $k$ admitting a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every $uv\in E$ is denoted by $χ'_Σ(G)$. It has been conjectured that $χ'_Σ(G)\leq Δ+ 2$ for every connected graph of order at least three different from the cycle $C_5$, where $Δ$ is the maximum degree of $G$. It is known that $χ'_Σ(G) = Δ+ O(Δ^\frac{5}{6}\ln^\frac{1}{6}Δ)$ for a graph $G$ without isolated edges. We improve this upper bound to $χ'_Σ(G) = Δ+ O(Δ^\frac{1}{2})$ using a simpler approach involving a combinatori…
A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $θ(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is distinguishing. We generalize this concept to edge-coloring by defining an alternative index $θ'(G)$. We consider $θ'$ for some families of graphs and find its connection with edge-cycles of the automorphism group. Then we show that $θ'(G)=2$ if and only if $G\simeq K_{1,2}$ and $θ'(G)=3$ if and only if $G\simeq P_4, K_{1,3}$ or $K_3$. Moreover, we prove some aux…
Symmetry Parameters of Two-Generator Circulant Graphs
本文研究由单顶点双环电压图导出的双生成元循环图，完整刻画其自同构群，并确定其确定数、区分数及2-区分代价，同时将结果推广到相应的细分图。
The Mycielskian construction, denoted $μ(G)$, takes a finite simple graph $G$ to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted $μ_t(G)$, takes $G$ to a larger graph with the same chromatic number but with larger odd girth. In this chapter we look at symmetry parameters of $μ(G)$ and $μ_t(G)$ in terms of the same parameters of $G$. These symmetry parameters include determining number, distinguishing number, and cost of distinguishing.
Symmetry Parameters of Praeger-Xu Graphs
本文计算了Praeger-Xu图的确定数与区分数，证明大多数此类图是2可区分的，并给出了相应的2区分代价。
A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by $χ\,'_s(G)$ called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds $χ\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or one …
The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, 1. We prove a lemma that may be considered a variant of the Motion lemma of \cite{RS} and use this to give examples of several families of graphs which satisfy $χ_D(G)=χ(G)+1$. 2.We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. We also desc…
Vertex-distinguishing edge coloring of graphs
本文研究了图的顶点区分边着色问题，证明了顶点区分色数的上界为floor(5.5k(G)+6.5)，并针对d-正则图给出了更优的上界。
Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice
本文在无选择公理的ZF集合论中证明若干关于无限局部有限连通图的色数、边色数、区分数与区分指数的命题等价于Kőnig引理，并给出与AC或Kőnig引理等价的覆盖与支配集存在性新条件。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。