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Total mutual-visibility in Hamming graphs

100%

Bujtás C. , Klavžar S. , Tian J.

Opuscula Mathematica

2025

tom Vol. 45, no. 1

63--78

If G is a graph and X ⊆ V (G), then X is a total mutual-visibility set if every pair of vertices x and y of G admits the shortest x, y-path P with V (P) ∩ X ⊆ {x, y}. The cardinality of the largest total mutual-visibility set of G is the total mutual-visibility number μt(G) of G. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values μt(Kn1 □Kn2 □Kn3 ) are determined. It is proved that μt(Kn1 □ · · · □Knr ) = O(Nr−2), where N = n1 + · · · + nr, and that μt(Ks□,r) = Θ(sr−2) for every r ≥ 3, where Ks□,r denotes the Cartesian product of r copies of Ks. The main theorems are also reformulated as Turán-type results on hypergraphs.

Radio Graceful Hamming Graphs

100%

Niedzialomski A.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 4

1007-1020

For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio graceful Hamming graphs. The main result shows that the Cartesian product of t copies of a complete graph is radio graceful for certain t. Graphs of this form provide infinitely many examples of radio graceful graphs of arbitrary diameter. We also show that these graphs are not radio graceful for large t.

Sierpiński graphs as spanning subgraphs of Hanoi graphs

100%

Hinz A. , Klavžar S. , Zemljič S.

Open Mathematics

2013

tom 11

nr 6

1153-1157

Hanoi graphs H pn model the Tower of Hanoi game with p pegs and n discs. Sierpinski graphs S pn arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that S pn embeds as a spanning subgraph into H pn if and only if p is odd or, trivially, if n = 1.