The square of every two-connected graph is Hamiltonian

Fleischner, Herbert

doi:10.1016/0095-8956(74)90091-4

Cited by 156 publications

(87 citation statements)

References 2 publications

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“…An end block of il block-path chain G is a block of G which contains at most one cut vertex of G. Following the terminology of [2], given a graph G, D(G) is the set of all edges of G which are not incident with vertices whose degree in G is two. A DT-subgraph of a graph G is a sub graph H of G such that every edge of H is incident with a vertex whose degree in G is two.…”

Section: Structure Of Minimal Blocksmentioning

confidence: 99%

A catalog of minimal blocks

Hobbs¹

1973

J. RES. NATL. BUR. STAN. SECT. B. MATH. SCI.

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In this paper we provide a catalog of the minimal blocks with 10 and fewer vertices, togel her with a discussion of the methods and theorems used to produce the catalog. In addition, we prove a theore m which is a strengthening of a similar theorem of Fleischner [2] on the structure of minimal blocks.Key words: Blocks; combinatorics; examples of graphs; graph theory; minimal blocks; planar graphs; thickness of graphs; two-connected graphs. Description of the CatalogThe majority of the definitions used here will be found in (3),1 with the terms "point", "line", and "cycle" replaced by vertex, edge, and circuit. In particular, a block is a connected graph with no cut vertex, and a block is minimal if no spanning subgraph of the block with fewe r edges is also a block. We consider the graph with one vertex and no edges (the vertex graph) and the graph with two vertices and a single edge joining them (the link graph) to be minimal blocks. To distinguish between a path p and the graph which contains exactly the edges and vertices of p, we denote the graph by Ipl; for simplicity of terminology, we will refer to both p and Ipl as "paths." If p is a path, Let A!, . .. , Ak be k disjoint graphs which are paths, with k ;;,; 2, such that path A i contains mi vertices, for each i E {I, ... , k}, with m, ;;,; m2 ;;,; ... ;;,; mk ;;,; 1. Let a and,B b e vertices not in., k}, let ai be a Hamiltonian path inAi. Let the graph P (ml, m2, where X(G) denotes the edge set of graph G) such that A" . . . , Ak are sub graphs of P. We call P(ml, . . . , mk) a partition graph; clearly a partition graph with n vertices is completely determined up to isomorphism by a partition of the integer n -2. Further, it is clear that each partition graph is a minimal block. In the sequence m" . . . , mk, if m s+, = m s+2 = . . . = m s+,. for some integer s and integer r;;'; 2, we may write m" . . . , mk in the form m" . .. m s , r X ms+l, ms+r+I, . . . , mk. For example, P(3, 4 X 2) is shown in figure 1. Using this notation, the catalog at the end of this paper gives all minimal blocks with 10 and fewer vertices_ Above each of the drawings of a graph with 7 or more vertices is a sequence in parentheses_ This sequence is the degree sequence of the associated graph, i_e_, the sequence of degrees of the vertices of the graph in descending order. The

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Section: Structure Of Minimal Blocksmentioning

confidence: 99%

A catalog of minimal blocks

Hobbs¹

1973

J. RES. NATL. BUR. STAN. SECT. B. MATH. SCI.

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“…Here, we briefly review those that are concerned with the squares and the cubes of graphs. It was shown that the square of every 2-connected graph is hamiltonian [10,12]. In fact, it is hamiltonian-connected and is 1-hamiltonian provided its order is at least four [4].…”

Section: Preliminariesmentioning

confidence: 99%

Single-source three-disjoint path covers in cubes of connected graphs

Park

Ihm

2013

Information Processing Letters

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A k-disjoint path cover (k-DPC for short) of a graph is a set of k internally vertex-disjoint paths from given sources to sinks that collectively cover every vertex in the graph. In this paper, we establish a necessary and sufficient condition for the cube of a connected graph to have a 3-DPC joining a single source to three sinks. We also show that the cube of a connected graph always has a 3-DPC joining arbitrary two vertices.

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“…Fleischner proved that the square of every 2-connected graph is hamiltonian [11] (for an alternative proof, refer to [13] or [22]). In fact, the square of a 2-connected graph is both hamiltonianconnected and 1-hamiltonian provided that its order is at least four [4].…”

Section: Strong Hamiltonian Propertiesmentioning

confidence: 99%

Disjoint path covers in cubes of connected graphs

Park

Ihm

2014

Discrete Mathematics

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Given a graph G, and two vertex sets S and T of size k each, a many-tomany k-disjoint path cover of G joining S and T is a collection of k disjoint paths between S and T that cover every vertex of G. It is classified as paired if each vertex of S must be joined to a designated vertex of T , or unpaired if there is no such constraint. In this article, we first present a necessary and sufficient condition for the cube of a connected graph to have a paired 2-disjoint path cover. Then, a corresponding condition for the unpaired type of 2-disjoint path cover problem is immediately derived. It is also shown that these results can easily be extended to determine if the cube of a connected graph has a hamiltonian path from a given vertex to another vertex that passes through a prescribed edge.

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The square of every two-connected graph is Hamiltonian

Cited by 156 publications

References 2 publications

A catalog of minimal blocks

A catalog of minimal blocks

Single-source three-disjoint path covers in cubes of connected graphs

Disjoint path covers in cubes of connected graphs

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