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 { 0,1,\textellipsis,n-1} 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={d1,d2}⊆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.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

Characterizing 2-distance graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Ramuel P. Ching ◽  
I. J. L. Garces
Keyword(s):  
Complete Graph ◽  
Distance Graph ◽  
Simple Graph ◽  
Distance Graphs ◽  
Vertex Set

Let [Formula: see text] be a finite simple graph. The [Formula: see text]-distance graph [Formula: see text] of [Formula: see text] is the graph with the same vertex set as [Formula: see text] and two vertices are adjacent if and only if their distance in [Formula: see text] is exactly [Formula: see text]. A graph [Formula: see text] is a [Formula: see text]-distance graph if there exists a graph [Formula: see text] such that [Formula: see text]. In this paper, we give three characterizations of [Formula: see text]-distance graphs, and find all graphs [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] is an integer, [Formula: see text] is the path of order [Formula: see text], and [Formula: see text] is the complete graph of order [Formula: see text].

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

scholarly journals Distance graphs with maximum chromatic number

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Javier Barajas ◽  
Oriol Serra
Keyword(s):  
Chromatic Number ◽  
Distance Graph ◽  
Distance Graphs ◽  
Maximum Value ◽  
Finite Set ◽  
International Audience

International audience Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.

scholarly journals Hamiltonicity of Minimum Distance Graphs of 1-Perfect Codes

10.37236/2158 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Alexander Mikhailovich Romanov
Keyword(s):  
Minimum Distance ◽  
Hamiltonian Cycle ◽  
Distance Graph ◽  
Perfect Code ◽  
Perfect Codes ◽  
Distance Graphs

A 1-perfect code $\mathcal{C}_{q}^{n}$ is called Hamiltonian if its minimum distance graph $G(\mathcal{C}_{q}^{n})$ contains a Hamiltonian cycle. In this paper, for  all admissible lengths $n \geq 13$, we construct   Hamiltonian nonlinear ternary 1-perfect  codes,   and for  all admissible lengths $n \geq 21$, we construct  Hamiltonian nonlinear quaternary 1-perfect  codes. The existence of Hamiltonian nonlinear $q$-ary 1-perfect  codes of length $N = qn + 1$ is reduced to the question of the existence of such codes of length $n$. Consequently,  for   $q = p^r$, where $p$ is prime, $r \geq 1$ there exist Hamiltonian nonlinear $q$-ary 1-perfect  codes of length $n = (q ^{m} -1) / (q-1)$, $m \geq 2$. If $q =2, 3, 4$, then $ m \neq 2$.  If $q =2$, then $ m \neq 3$.

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 On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Olivier Baudon ◽  
Julien Bensmail ◽  
Rafał Kalinowski ◽  
Antoni Marczyk ◽  
Jakub Przybyło ◽  
...  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Vertex Set ◽  
International Audience ◽  
Positive Integers

Graph Theory International audience A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ &#x007b;1,\textellipsis,k&#x007d;, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.

scholarly journals The generalized 3-connectivity of Cartesian product graphs

2012 ◽  
Vol Vol. 14 no. 1 (Graph Theory) ◽  
Author(s):  
Hengzhe Li ◽  
Xueliang Li ◽  
Yuefang Sun
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Cartesian Product ◽  
Vertex Connectivity ◽  
Connected Graphs ◽  
Generalized Connectivity ◽  
Product Graphs ◽  
International Audience ◽  
Cartesian Product Graphs ◽  
Internally Disjoint Trees

Graph Theory International audience The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection \T-1, T (2), ... , T-r\ of trees in G is said to be internally disjoint trees connecting S if E(T-i) boolean AND E(T-j) - empty set and V (T-i) boolean AND V(T-j) = S for any pair of distinct integers i, j, where 1 <= i, j <= r. For an integer k with 2 <= k <= n, the k-connectivity kappa(k) (G) of G is the greatest positive integer r for which G contains at least r internally disjoint trees connecting S for any set S of k vertices of G. Obviously, kappa(2)(G) = kappa(G) is the connectivity of G. Sabidussi's Theorem showed that kappa(G square H) >= kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.

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