Call a colouring of a graph \emph{distinguishing} if the only automorphism of this graph which preserves said colouring is the identity. Let $H$ be an arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not contain an induced subgraph isomorphic to $H$. Kargul, Musiał, Pal and Gorzkowska showed that if $n$ is a natural number greater than two, then every finite connected $K_{1,n}$-free graph of order at least six admits a distinguishing edge colouring with at most $n-1$ colours. We extend this result to all locally finite connected $K_{1,n}$-free graphs of order at least six.
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 index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $α$-regular graph for some infinite cardinal $α$ then $D'(G) \le 2$, proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if $G$ is a graph with infinite minimum degree and at most $2^α$ vertices of degree $α$ for every infinite cardinal $α$, then $D'(G) \le 3$. In particular, $D'(G) \le 3$ if $G$ has infinite minimum degree and order at most $2^{\aleph_0}$.
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing edge-colouring with two colours. We show that all such graphs except $K_2$ admit a distinguishing edge-colouring with three colours. This result also extends to infinite, locally finite graphs. Furthermore, we are able to show that there are arbitrary large infinite cardinals $κ$ such that every connected $κ$-regular graph has distinguishing edge-colouring with two colo…
图$G$的区别数$D(G)$是对图$G$的顶点着色所需的最小颜色数，使得唯一的颜色保持自同构是单位元。对于无限图，$D(G)$是有界的价的上确界，而对于有限图，$Δ(G)+1$，其中$Δ(G)$是最大价。给定一个有限或无限树$T$，其有界有限价$k$和一个整数$c$，其中$2 \leq c \leq k$，我们感兴趣的是用$c$颜色对$T$的顶点着色，使得每个保色自同构固定尽可能多的顶点.在这个意义上，我们表明，总是存在一个$c$染色，所有的顶点，其距离下一个叶至少是$\lceil\log_ck\rceil$是固定的任何颜色保持自同构，在许多情况下，可以做得更好。
Extending the well-known star-comb lemma for infinite graphs, we characterise the graphs that do not contain an infinite comb or an infinite star, respectively, attached to a given set of vertices. We offer several characterisations: in terms of normal trees, tree-decompositions, ranks of rayless graphs and tangle-distinguishing separators.
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…
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…
Except for Koshy who devotes seven pages to applications of Fibonacci Numbers to electric circuits, most books and the Fibonacci Quarterly have been relatively silent on applications of graphs and electric circuits to Fibonacci numbers. This paper continues a recent trend of papers studying the interplay of graphs, circuits, and Fibonacci numbers by presenting and studying the Circuit Array, an infinite 2-dimensional array whose entries are electric resistances labelling edge values of circuits associated with a family of graphs. The Circuit Array has several features distinguishing it from ot…
We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-free graphs are finite cographs. We also prove that $N$-free chain complete posets and $N$-free posets with no infinite antichains are series-parallel. As a consequence, we obtain that every $N$-free poset with no infinite antichain has a chain and a partition into antichains so that each part intersects the chain. This answers a conjecture of Aharoni and Korman (Order \textbf{9} (1992) 245--253) in this case.
