On the connectivity and restricted edge-connectivity of 3-arc graphs

Balbuena, C.; García–Vázquez, P.; Montejano, Luis

doi:10.1016/j.dam.2013.08.010

Cited by 5 publications

(5 citation statements)

References 21 publications

(25 reference statements)

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“…The Hamiltonian problems and related topics are classical in combinatorics and have been received extensive treatment. See [ 1 – 3 ] for more results on this area of investigation.…”

Section: Introductionmentioning

confidence: 99%

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On Hamiltonian Decomposition Problem of 3-Arc Graphs

Sun

Liang

2022

Computational Intelligence and Neuroscience

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A 4-tuple y , x , v , w in a graph is a 3-arc if each of y , x , v and x , v , w is a path. The 3-arc graph of H is the graph with vertex set all arcs of H and edge set containing all edges joining x y and v w whenever y , x , v , w is a 3-arc of H . A Hamilton cycle is a closed path meeting each vertex of a graph. A graph H including a Hamilton cycle is called Hamiltonian and H has a Hamiltonian decomposition provided its edge set admits a partition into disjoint Hamilton cycles (possibly with a single perfect matching). The current paper proves that every connected 3-arc graph consists of more than one Hamilton cycle. Since the 3-arc graph of a cubic graph is 4-regular, it further proves that each 3-arc graph of a cubic graph in a certain family has a Hamiltonian decomposition.

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“…The Hamiltonian problems and related topics are classical in combinatorics and have been received extensive treatment. See [ 1 – 3 ] for more results on this area of investigation.…”

Section: Introductionmentioning

confidence: 99%

“…We refer to [ 1 ] for terms and notation not mentioned. The degree of u in a graph H is written as d ( u ), and the minimum degree of H is denoted by δ ( H ).…”

Section: Introductionmentioning

confidence: 99%

On Hamiltonian Decomposition Problem of 3-Arc Graphs

Sun

Liang

2022

Computational Intelligence and Neuroscience

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“…On the other hand, the quotient graph of X(G) with respect to the partition P = {{uv, vu} : {u, v} ∈ E(G)} of A(G) is isomorphic to the graph obtained from the square of L(G) by deleting the edges of L(G). The reader is referred to [14,13,2] respectively for results on the diameter and connectivity, the independence, domination and chromatic numbers, and the edge-connectivity and restricted edge-connectivity of 3-arc graphs.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonicity of 3-Arc Graphs

Zhou

2013

Graphs and Combinatorics

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An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. We prove that any connected 3-arc graph is hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are hamiltonian. As a corollary we obtain that any vertex-transitive graph which is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three must be hamiltonian. This confirms the conjecture, for this family of vertex-transitive graphs, that all vertex-transitive graphs with finitely many exceptions are hamiltonian. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.

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“…. Broersma and Hoede [9] defined in general path graphs P k (G) of G for any positive integer k as follows: P k (G) has for its vertex-set the set P k G of all distinct paths in G having k vertices, and two vertices in P k (G) are adjacent if they represent two paths P, Q ∈ P k G whose union forms either a path P k+1 or a cycle C k in G. Some improvement of their paper was subsequently given by [27,7,28].…”

Section: Introductionmentioning

confidence: 99%

“…As in [9] (also see, [7]) by a sigraph S we mean a 2 graph G = (V, E) called the underlying graph of S and denoted by S u , in which each edge x carries a value s(x) ∈ {+1, −1} called its sign; an edge x is positive or negative according to whether s(x) = +1 or s(x) = −1. The set of positive edges of S is denoted by E + (S) and E − (S) = E(G) − E + (S) is the set of negative edges of S. Graphs themselves regarded as sigraphs in which every edge is positive.…”

Section: Introductionmentioning

confidence: 99%

An Optimal Algorithm to Detect Balancing in Common-edge Sigraph

Sinha¹,

Sethi²

2014

IJCA

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A signed graph (or sigraph in short) S is a graph G in which each edge x carries a value s(x) ∈ {+1, −1} called its sign denoted specially as S = (G, s). Given a sigraph S, a new sigraph C E (S), called the common-edge sigraph of S is that sigraph whose vertex-set is the set of pairs of adjacent edges in S and two vertices of C E (S) are adjacent if the corresponding pairs of adjacent edges of S have exactly one edge in common, and the sign of the edge is the sign of the common edge. If all the edges of the sigraph S carry + sign then S is actually a graph and the corresponding commonedge sigraph is termed as the common-edge graph. In this paper, algorithms are defined to obtain a common-edge sigraph and detect whether it is balanced or not in O(n 3 ) steps which will be optimal in nature.

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On the connectivity and restricted edge-connectivity of 3-arc graphs

Abstract: A 3 − arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G.

Cited by 5 publications

References 21 publications

On Hamiltonian Decomposition Problem of 3-Arc Graphs

On Hamiltonian Decomposition Problem of 3-Arc Graphs

Hamiltonicity of 3-Arc Graphs

An Optimal Algorithm to Detect Balancing in Common-edge Sigraph

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