Decomposing the complete graph into Hamiltonian paths  (cycles)  and 3-stars

Marco Buratti has conjectured that, given an odd prime $p$ and a multiset $L$ containing $p-1$ integers taken from $\{1,\ldots,(p-1)/2\}$, there exists a Hamiltonian path in the complete graph with $p$ vertices whose multiset of edge-lengths is equal to $L$ modulo $p$. We give a positive answer to this conjecture in the case of multisets of the type $\{1^a,2^b,3^c\}$ by completely classifying such multisets that are linearly or cyclically realizable.

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On the Buratti-Horak-Rosa Conjecture about Hamiltonian Paths in Complete Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3879 ◽

2014 ◽

Vol 21 (2) ◽

Cited By ~ 1

Author(s):

Anita Pasotti ◽

Marco Antonio Pellegrini

Keyword(s):

Positive Integer ◽

Complete Graph ◽

Complete Solution ◽

Complete Graphs ◽

Hamiltonian Paths

In this paper we investigate a problem proposed by Marco Buratti, Peter Horak and Alex Rosa (denoted by BHR-problem) concerning Hamiltonian paths in the complete graph with prescribed edge-lengths. In particular we solve BHR$(\{1^a, 2^b, t^c\})$ for any even integer $t \geq 4$, provided that $a+b \geq t-1$. Furthermore, for $t=4, 6, 8$ we present a complete solution of BHR$(\{ 1^a,2^b,t^c \})$ for any positive integer $a,b,c$.

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Absolute Differences Along Hamiltonian Paths

The Electronic Journal of Combinatorics ◽

10.37236/5159 ◽

2015 ◽

Vol 22 (3) ◽

Author(s):

Francesco Monopoli

Keyword(s):

Arithmetic Progression ◽

Complete Graph ◽

Hamiltonian Cycle ◽

Hamiltonian Path ◽

Extreme Case ◽

Hamiltonian Paths ◽

Additive Energy ◽

The Absolute ◽

Small Additive

We prove that if the vertices of a complete graph are labeled with the elements of an arithmetic progression, then for any given vertex there is a Hamiltonian path starting at this vertex such that the absolute values of the differences of consecutive vertices along the path are pairwise distinct. In another extreme case where the label set has small additive energy, we show that the graph actually possesses a Hamiltonian cycle with the property just mentioned. These results partially solve a conjecture by Z.-W. Sun.

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On the Generalized Distance Energy of Graphs

Mathematics ◽

10.3390/math8010017 ◽

2019 ◽

Vol 8 (1) ◽

pp. 17 ◽

Cited By ~ 4

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Connected Graph ◽

Distance Matrix ◽

Wiener Index ◽

Diagonal Matrix ◽

Upper And Lower Bounds ◽

Star Graph ◽

Extremal Graphs ◽

Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽

10.3390/math9050512 ◽

2021 ◽

Vol 9 (5) ◽

pp. 512

Author(s):

Maryam Baghipur ◽

Modjtaba Ghorbani ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Lower Bounds ◽

Complete Graph ◽

Distance Matrix ◽

Upper And Lower Bounds ◽

Laplacian Eigenvalue ◽

Signless Laplacian ◽

Graph Parameters ◽

Simple Connected Graph ◽

Second Largest Eigenvalue ◽

Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

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Undirected Complete Graph to Design New Public Key Cryptosystem

Journal of Physics Conference Series ◽

10.1088/1742-6596/1897/1/012045 ◽

2021 ◽

Vol 1897 (1) ◽

pp. 012045

Author(s):

Karrar Taher R. Aljamaly ◽

Ruma Kareem K. Ajeena

Keyword(s):

Complete Graph ◽

Public Key ◽

Public Key Cryptosystem ◽

New Public

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Turán theorems for unavoidable patterns

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500412100027x ◽

2021 ◽

pp. 1-20

Author(s):

ANTÓNIO GIRÃO ◽

BHARGAV NARAYANAN

Keyword(s):

Complete Graph ◽

Blow Up ◽

Ramsey Numbers ◽

Order Of Magnitude ◽

Transitive Tournament ◽

Unavoidable Patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

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