In 2009, Adamus showed that if $G$ is a balanced tripartite graph of order $3n$, $n \geq 2$, with at least $3n^2 - 2n + 2$ edges, then $G$ is hamiltonian and, in fact, $G$ is pancyclic. Removing all but one edge incident with any vertex... more
In 2009, Adamus showed that if $G$ is a balanced tripartite graph of order $3n$, $n \geq 2$, with at least $3n^2 - 2n + 2$ edges, then $G$ is hamiltonian and, in fact, $G$ is pancyclic. Removing all but one edge incident with any vertex of the complete, balanced tripartite graph $K(n,n,n)$ shows that this result is best possible. Here we extend the result to balanced $k$-partite graphs of order $kn$. We prove that for all integers $k\geq 3$ and $n\geq 1$, every balanced $k$-partite graph with $kn$ vertices and at least ${{(k^2-k)n^2-2n(k-1)+4}\over 2}$ edges is pancyclic. We also prove a similar result for $k$-partite graphs that are not balanced.
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The girth of a graph with a Hamiltonian cycle and t chords is investigated. In particular, for any integer t>0 let g(t) denote the smallest number such that any Hamiltonian graph G with n vertices and n+t edges has girth at most... more
The girth of a graph with a Hamiltonian cycle and t chords is investigated. In particular, for any integer t>0 let g(t) denote the smallest number such that any Hamiltonian graph G with n vertices and n+t edges has girth at most g(t)n+c, where c is a constant independent of n. It is shown that there exist constants c 1 and c 2 such that (c 1 (logt))/t≤g(t)≤(c 2 (logt)/t). For small values of t (1≤t≤8), g(t) is determined precisely.
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ABSTRACT We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number... more
ABSTRACT We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a graph is sufficient to imply certain hamiltonian properties in graphs. For K1,m-free grphs we obtain generalizations of known results. In particular we show:Theorem. Let r ≥ 1 and m ≥ 3 be integers. Then for each nonnegative function f(r, m) there exists a constant C = C(r, m, f(r, m)) such that if G is a graph of order n (n ≥ r, n > m) with δr(G) ≥ (n/3) + C and β (G) ≥ f(r, m), then (a) G is traceable if δ(G) ≥ r and G is connected; (b) G is hamiltonian if δ(G) ≥ r + 1 and G is 2-connected; (c) G is hamiltonian-connected if δ(G) ≥ r + 2 and G is 3-connected. © 1995 John Wiley & Sons, Inc.
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The eccentricitye(v) of a vertexv of a connected graphG is the maximum distance fromv among the vertices ofG. A nondecreasing sequencea 1,a 2, ...,a p of nonnegative integers is said to be an eccentric sequence if there exists a connected... more
The eccentricitye(v) of a vertexv of a connected graphG is the maximum distance fromv among the vertices ofG. A nondecreasing sequencea 1,a 2, ...,a p of nonnegative integers is said to be an eccentric sequence if there exists a connected graphG of orderp whose vertices can be labelledv 1,v 2, ...,v p so thate(v i )=a i for alli. Several properties of eccentric sequences are exhibited, and a necessary and sufficient condition for a sequence to be eccentric is presented. Sequences which are the eccentricity sequences of trees are characterized. Some properties of the eccentricity sequences of self-complementary graphs are obtained. It is shown that the radius of a nontrivial self-complementary graph is two.
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... We define mp(G) = 0 if G has neither a perfect matching nor an almost-perfect matching. This concept of matching preclusion was introduced by Brigham et al. [3] and further studied by Cheng and Lipták [5] and Cheng et al. ...
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The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the... more
The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the minimum order of such a digraphD is determined. Also, given two finite sequences of nonnegative integers, a necessary and sufficient condition is provided for which these sequences are the outdegree sequences of the two sets of an asymmetric bipartite digraph.
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An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by... more
An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by a study of subgraphs and supergraphs of Eulerian graphs. After an introduction to randomly Eulerian graphs and digraphs, we conclude with a summary of a variety of results involving enumeration.
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The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the... more
The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the minimum order of such a digraphD is determined. Also, given two finite sequences of nonnegative integers, a necessary and sufficient condition is provided for which these sequences are the outdegree sequences of the two sets of an asymmetric bipartite digraph.
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An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by... more
An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by a study of subgraphs and supergraphs of Eulerian graphs. After an introduction to randomly Eulerian graphs and digraphs, we conclude with a summary of a variety of results involving enumeration.
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The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the... more
The degree setD D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD D =S. Furthermore, the minimum order of such a digraphD is determined. Also, given two finite sequences of nonnegative integers, a necessary and sufficient condition is provided for which these sequences are the outdegree sequences of the two sets of an asymmetric bipartite digraph.
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An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by... more
An overview of Eulerian graphs is presented. In particular, characterizations of Eulerian graphs and digraphs as well as algorithms for constructing Eulerian circuits are discussed. A solution to the Chinese postman problem is followed by a study of subgraphs and supergraphs of Eulerian graphs. After an introduction to randomly Eulerian graphs and digraphs, we conclude with a summary of a variety of results involving enumeration.
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In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore... more
In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where . In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with .
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... On k-ordered graphs. Jill R. Faudree 1,* ,; Ralph J. Faudree 2 ,; Ronald J. Gould 3 ,; Michael S. Jacobson 4 ,; Linda Lesniak 5. Article first published online: 8 SEP 2000. DOI:... more
... On k-ordered graphs. Jill R. Faudree 1,* ,; Ralph J. Faudree 2 ,; Ronald J. Gould 3 ,; Michael S. Jacobson 4 ,; Linda Lesniak 5. Article first published online: 8 SEP 2000. DOI: 10.1002/1097-0118(200010)35:2<69::AID-JGT1>3.0.CO;2-I. Copyright © 2000 John Wiley & ...
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... On k-ordered graphs. Jill R. Faudree 1,* ,; Ralph J. Faudree 2 ,; Ronald J. Gould 3 ,; Michael S. Jacobson 4 ,; Linda Lesniak 5. Article first published online: 8 SEP 2000. DOI:... more
... On k-ordered graphs. Jill R. Faudree 1,* ,; Ralph J. Faudree 2 ,; Ronald J. Gould 3 ,; Michael S. Jacobson 4 ,; Linda Lesniak 5. Article first published online: 8 SEP 2000. DOI: 10.1002/1097-0118(200010)35:2<69::AID-JGT1>3.0.CO;2-I. Copyright © 2000 John Wiley & ...
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ABSTRACT For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore&#39;s... more
ABSTRACT For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore&#39;s classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k. © 1997 John Wiley &amp; Sons, Inc.
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We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a... more
We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a graph is sufficient to imply certain hamiltonian properties in graphs. For K1,m-free grphs we obtain generalizations of known results. In particular we show:Theorem. Let r ≥ 1 and m ≥ 3 be integers. Then for each nonnegative function f(r, m) there exists a constant C = C(r, m, f(r, m)) such that if G is a graph of order n (n ≥ r, n > m) with δr(G) ≥ (n/3) + C and β (G) ≥ f(r, m), then (a) G is traceable if δ(G) ≥ r and G is connected; (b) G is hamiltonian if δ(G) ≥ r + 1 and G is 2-connected; (c) G is hamiltonian-connected if δ(G) ≥ r + 2 and G is 3-connected. © 1995 John Wiley & Sons, Inc.
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ABSTRACT For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore&#39;s... more
ABSTRACT For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ore&#39;s classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k. © 1997 John Wiley &amp; Sons, Inc.
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For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of... more
For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 ≤ k ≤ n/2, and deg(u) + deg(v) ≥ n + (3k − 9)/2 for every pair u, v of nonadjacent vertices of G, then G is k-ordered hamiltonian. Minimum degree conditions are also given for k-ordered hamiltonicity. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 199–210, 2003