scholarly journals Absolute Differences Along Hamiltonian Paths

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.

scholarly journals Hamiltonian paths in odd graphs

2009 ◽  
Vol 3 (2) ◽  
pp. 386-394 ◽  
Author(s):  
Letícia Bueno ◽  
Luerbio Faria ◽  
Figueiredo De ◽  
Fonseca Da
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Transitive Graph ◽  
Direct Computation ◽  
Hamiltonian Paths ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs ◽  
Connected Vertex

Lov?sz conjectured that every connected vertex-transitive graph has a Hamiltonian path. The odd graphs Ok form a well-studied family of connected, k-regular, vertex-transitive graphs. It was previously known that Ok has Hamiltonian paths for k ? 14. A direct computation of Hamiltonian paths in Ok is not feasible for large values of k, because Ok has (2k - 1, k - 1) vertices and k/2 (2k - 1, k - 1) edges. We show that Ok has Hamiltonian paths for 15 ? k ? 18. Instead of directly running any heuristics, we use existing results on the middle levels problem, therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph. We show that further improved results for the middle levels problem can be used to find Hamiltonian paths in Ok for larger values of k.

MUTUALLY INDEPENDENT HAMILTONIAN PATHS AND CYCLES IN HYPERCUBES

2006 ◽  
Vol 07 (02) ◽  
pp. 235-255 ◽  
Author(s):  
CHAO-MING SUN ◽  
CHENG-KUAN LIN ◽  
HUA-MIN HUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Hamiltonian Cycles ◽  
Hamiltonian Paths ◽  
Mutually Independent ◽  
Hamiltonian Laceability

Two hamiltonian paths P1 = 〈v1, v2, …, vn(G) 〉 and P2 = 〈 u1, u2, …, un(G) 〉 of G are independent if v1 = u1, vn(G) = un(G), and vi ≠ ui for 1 < i < n(G). A set of hamiltonian paths {P1, P2, …, Pk} of G are mutually independent if any two different hamiltonian paths in the set are independent. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two nodes from different partite sets. A bipartite graph is k-mutually independent hamiltonian laceable if there exist k-mutually independent hamiltonian paths between any two nodes from different partite sets. The mutually independent hamiltonian laceability of bipartite graph G, IHPL(G), is the maximum integer k such that G is k-mutually independent hamiltonian laceable. Let Qn be the n-dimensional hypercube. We prove that IHPL(Qn) = 1 if n ∈ {1,2,3}, and IHPL(Qn) = n - 1 if n ≥ 4. A hamiltonian cycle C of G is described as 〈 u1, u2, …, un(G), u1 〉 to emphasize the order of nodes in C. Thus, u1 is the beginning node and ui is the i-th node in C. Two hamiltonian cycles of G beginning at u, C1 = 〈 v1, v2, …, vn(G), v1 〉 and C2 = 〈 u1, u2, …, un(G), u1 〉, are independent if u = v1 = u1, and vi ≠ ui for 1 < i ≤ n(G). A set of hamiltonian cycles {C1, C2, …, Ck} of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonianicity of graph G, IHC(G), is the maximum integer k such that for any node u of G there exist k-mutually independent hamiltonian cycles of G starting at u. We prove that IHC(Qn) = n - 1 if n ∈ {1,2,3} and IHC(Qn) = n if n ≥ 4.

scholarly journals Hamiltonian Paths in the Complete Graph with Edge-Lengths 1, 2, 3

10.37236/316 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Stefano Capparelli ◽  
Alberto Del Fra
Keyword(s):  
Complete Graph ◽  
Hamiltonian Path ◽  
Positive Answer ◽  
Hamiltonian Paths ◽  
Modulo P

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.

Hamiltonian Cycles in Products of Graphs

1975 ◽  
Vol 17 (5) ◽  
pp. 763-765 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Complete Bipartite Graph ◽  
Hamiltonian Path ◽  
Hamiltonian Cycles ◽  
Vertex Set ◽  
Cycle Path

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.

scholarly journals On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Christian Löwenstein ◽  
Dieter Rautenbach ◽  
Roman Soták
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Large Distance ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Distance Graph ◽  
Hamiltonian Paths ◽  
Distance Graphs ◽  
Vertex Set ◽  
International Audience

Graph Theory International audience For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set &#x007b; 0,1,\textellipsis,n-1&#x007d; and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D=&#x007b;d1,d2&#x007d;⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.

A Parallel Reduction of Hamiltonian Cycle to Hamiltonian Path in Tournaments

1995 ◽  
Vol 19 (3) ◽  
pp. 432-440 ◽  
Author(s):  
E. Bampis ◽  
M. Elhaddad ◽  
Y. Manoussakis ◽  
M. Santha
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Parallel Reduction

scholarly journals On Cayley Digraphs That Do Not Have Hamiltonian Paths

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Dave Witte Morris
Keyword(s):  
Finite Group ◽  
Hamiltonian Path ◽  
Infinite Family ◽  
Hamiltonian Paths ◽  
Cayley Digraph

We construct an infinite family {Cay→(Gi;ai;bi)} of connected, 2-generated Cayley digraphs that do not have hamiltonian paths, such that the orders of the generators ai and bi are unbounded. We also prove that if G is any finite group with |[G,G]|≤3, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]|=4 or 5).

scholarly journals New Sufficient Conditions for Hamiltonian Paths

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
M. Sohel Rahman ◽  
M. Kaykobad ◽  
Jesun Sahariar Firoz
Keyword(s):  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Hamiltonian Path Problem

A Hamiltonian path in a graph is a path involving all the vertices of the graph. In this paper, we revisit the famous Hamiltonian path problem and present new sufficient conditions for the existence of a Hamiltonian path in a graph.

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