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1
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Maximum Independent Sets in Direct Products of Cycles or Trees with Arbitrary Graphs

100%
Paj T., Špacapan S.
Discussiones Mathematicae Graph Theory
|
2015
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tom 35
|
nr 4
675-688
EN
The direct product of graphs G = (V (G),E(G)) and H = (V (H),E(H)) is the graph, denoted as G×H, with vertex set V (G×H) = V (G)×V (H), where vertices (x1, y1) and (x2, y2) are adjacent in G × H if x1x2 ∈ E(G) and y1y2 ∈ E(H). Let n be odd and m even. We prove that every maximum independent set in Pn×G, respectively Cm×G, is of the form (A×C)∪(B× D), where C and D are nonadjacent in G, and A∪B is the bipartition of Pn respectively Cm. We also give a characterization of maximum independent subsets of Pn × G for every even n and discuss the structure of maximum independent sets in T × G where T is a tree.
2
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Isomorphisms of direct products of lattice-ordered groups

80%
Jakubík J.
Discussiones Mathematicae - General Algebra and Applications
|
2004
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tom 24
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nr 1
43-52
EN
In this paper we investigate sufficient conditions for the validity of certain implications concerning direct products of lattice-ordered groups.
3
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2-Tone Colorings in Graph Products

80%
Loe J., Middelbrooks D., Morris A., Wash K.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 1
55-72
EN
A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number for the direct product G×H, the Cartesian product G□H, and the strong product G⊠H.
4
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On acyclic colorings of direct products

80%
Špacapan S., Horvat A.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 2
323-333
EN
A coloring of a graph G is an acyclic coloring if the union of any two color classes induces a forest. It is proved that the acyclic chromatic number of direct product of two trees T₁ and T₂ equals min{Δ(T₁) + 1, Δ(T₂) + 1}. We also prove that the acyclic chromatic number of direct product of two complete graphs Kₘ and Kₙ is mn-m-2, where m ≥ n ≥ 4. Several bounds for the acyclic chromatic number of direct products are given and in connection to this some questions are raised.
5
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Adjacent vertex distinguishing edge colorings of the direct product of a regular graph by a path or a cycle

80%
Frigerio L., Lastaria F., Salvi N.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 3
547-557
EN
In this paper we investigate the minimum number of colors required for a proper edge coloring of a finite, undirected, regular graph G in which no two adjacent vertices are incident to edges colored with the same set of colors. In particular, we study this parameter in relation to the direct product of G by a path or a cycle.
6
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Isomorphic components of direct products of bipartite graphs

80%
Hammack R.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
231-248
EN
A standard result states the direct product of two connected bipartite graphs has exactly two components. Jha, Klavžar and Zmazek proved that if one of the factors admits an automorphism that interchanges partite sets, then the components are isomorphic. They conjectured the converse to be true. We prove the converse holds if the factors are square-free. Further, we present a matrix-theoretic conjecture that, if proved, would prove the general case of the converse; if refuted, it would produce a counterexample.
7
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A new proof of a theorem of Balcerzyk, Białynicki-Birula and Łoś

70%
O'Neill J. D.
Colloquium Mathematicum
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1996
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tom 70
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nr 2
191-194
8
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Some results on total domination in direct products of graphs

70%
Dorbec P., Gravier S., Klavžar S., Spacapan S.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 1
103-112
EN
Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.
9
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Median and quasi-median direct products of graphs

70%
Brešar B., Jha P., Klavžar S., Zmazek B.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 1-2
183-196
EN
Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G×P₃ is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.
10
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IF-filters of pseudo-BL-algebras

61%
Wojciechowska-Rysiawa M.
Discussiones Mathematicae - General Algebra and Applications
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2015
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tom 35
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nr 2
177-193
EN
Characterizations of IF-filters of a pseudo-BL-algebra are established. Some related properties are investigated. The notation of prime IF- filters and a characterization of a pseudo-BL-chain are given. Homomorphisms of IF-filters and direct product of IF-filters are studied.
11
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Union of Distance Magic Graphs

51%
Cichacz S., Nikodem M.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
239-249
EN
A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ from V to the set {1, . . . , n} such that the weight w(x) = ∑y∈NG(x) ℓ(y) of every vertex x ∈ V is equal to the same element μ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.
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