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1
Content available remote

End-regular and End-orthodox generalized lexicographic products of bipartite graphs

100%
Gu R., Hou H.
Open Mathematics
|
2016
|
tom 14
|
nr 1
229-236
EN
A graph X is said to be End-regular (End-orthodox) if its endomorphism monoid End(X) is a regular (orthodox) semigroup. In this paper, we determine the End-regular and the End-orthodox generalized lexicographic products of bipartite graphs.
2
Content available remote

Note on group distance magic complete bipartite graphs

81%
Cichacz S.
Open Mathematics
|
2014
|
tom 12
|
nr 3
529-533
EN
A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).
3
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On Cayley graphs of completely 0-simple semigroups

81%
Wang S., Li Y.
Open Mathematics
|
2013
|
tom 11
|
nr 5
924-930
EN
We give necessary and sufficient conditions for various vertex-transitivity of Cayley graphs of the class of completely 0-simple semigroups and its several subclasses. Moreover, the question when the Cayley graphs of completely 0-simple semigroups are undirected is considered.
4
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The join of split graphs whose completely regular endomorphisms form a monoid

81%
Hou H., Song Y., Gu R.
Open Mathematics
|
2017
|
tom 15
|
nr 1
833-839
EN
In this paper, completely regular endomorphisms of the join of split graphs are investigated. We give conditions under which all completely regular endomorphisms of the join of two split graphs form a monoid.
5
Content available remote

Chromatic Properties of the Pancake Graphs

62%
Konstantinova E.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
777-787
EN
Chromatic properties of the Pancake graphs Pn, n ⩾ 2, that are Cayley graphs on the symmetric group Symn generated by prefix-reversals are investigated in the paper. It is proved that for any n ⩾ 3 the total chromatic number of Pn is n, and it is shown that the chromatic index of Pn is n − 1. We present upper bounds on the chromatic number of the Pancake graphs Pn, which improve Brooks’ bound for n ⩾ 7 and Katlin’s bound for n ⩽ 28. Algorithms of a total n-coloring and a proper (n − 1)-coloring are given.
6
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Distinguishing Cartesian Products of Countable Graphs

62%
Estaji E., Imrich W., Kalinowski R., Pilśniak M., Tucker T.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 1
155-164
EN
The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.
7
Content available remote

Perfect codes in power graphs of finite groups

62%
Ma X., Fu R., Lu X., Guo M., Zhao Z.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1440-1449
EN
The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.
8
Content available remote

Onθ-commutators and the corresponding non-commuting graphs

52%
Shalchi S., Erfanian A., Farrokhi DG M.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1530-1538
EN
The θ-commutators of elements of a group with respect to an automorphism are introduced and their properties are investigated. Also, corresponding to θ-commutators, we define the θ-non-commuting graphs of groups and study their correlations with other notions. Furthermore, we study independent sets in θ-non-commuting graphs, which enable us to evaluate the chromatic number of such graphs.
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