The Cost of 2-Distinguishing Selected Kneser Graphs and Hypercubes
نویسنده
چکیده
A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. The minimum size of a label class in such a labeling of G is called the cost of 2-distinguishing and is denoted by ρ(G). This paper shows that ρ(K2m−1:2m−1−1) = m+1 – the only result so far on the cost of 2-distinguishing Kneser graphs. The result for Kneser graphs is adapted to show that ρ(Q2m−2) = ρ(Q2m−1) = ρ(Q2m) = m + 2 – a significant improvement on previously known bounds for the cost of 2-distinguishing
منابع مشابه
On Distinguishing and Distinguishing Chromatic Numbers of Hypercubes
The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number χD(G) of G. Extending these concepts to infinite graphs we prove that D(Qא0) = 2 and χD(Qא0) = 3, where Qא0 denotes the hypercube of cou...
متن کاملThe Distinguishing Chromatic Number of Kneser Graphs
A labeling f : V (G) → {1, 2, . . . , d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by χD(G), is the minimum d such that G has a proper d-distinguishing labeling. Let χ(G) be the chromatic nu...
متن کاملThe distinguishing chromatic number of bipartite graphs of girth at least six
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. We prove that if $G$ is a bipartite graph of girth at least six with the maximum ...
متن کاملShifts of the stable Kneser graphs and hom-idempotence
A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn. Larose et al. (1998) proved that Kneser graphs KG(n, k) are not weakly hom-idempotent for n ≥ 2k + 1, k ≥ 2. For s ≥ 2, we characterize all the shifts (i.e., automorphisms of the graph that map every vertex to one of its neighbors) ...
متن کاملSymmetries of the Stable Kneser Graphs
It is well known that the automorphism group of the Kneser graph KGn,k is the symmetric group on n letters. For n ≥ 2k + 1, k ≥ 2, we prove that the automorphism group of the stable Kneser graph SGn,k is the dihedral group of order 2n. Let [n] := [1, 2, 3, . . . , n]. For each n ≥ 2k, n, k ∈ {1, 2, 3, . . .}, the Kneser graph KGn,k has as vertices the k-subsets of [n] with edges defined by disj...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012