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Decomposition of the Complete rGraph into Complete rPartite rGraphs
 GRAPHS AND COMBINATORICS
, 1986
"... For n> r> 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and Pollak showed that fz(n) = n 1. Here we observe that f3(n) = n 2 and show that for every fixed r> 2, ..."
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Cited by 11 (3 self)
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For n> r> 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and Pollak showed that fz(n) = n 1. Here we observe that f3(n) = n 2 and show that for every fixed r> 2
Graphs and Combinatorics © SpringerVerlag 1986 Decomposition of the Complete rGraph into Complete rPartite rGraphs*
"... Abstract. For n> r> 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and Pollak showed that fz(n) = n 1. Here we observe that f3(n) = n 2 and show that for every fixed ..."
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Abstract. For n> r> 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete rgraph on n vertices into q complete rpartite rgraphs. Graham and Pollak showed that fz(n) = n 1. Here we observe that f3(n) = n 2 and show that for every fixed
Complete rpartite subgraphs of dense rgraphs, preprint
 Proc. London
"... Let r ≥ 3 and (ln n) −1/(r−1) ≤ α ≤ r −3. We show that: Every runiform graph on n vertices with at least αnr ⌊ /r! edges contains a complete rpartite graph with r − 1 parts of size α (ln n) 1/(r−1)⌋ and one part of size n1−αr−2 ⌉. This result follows from a more general digraph version: Let U1,.. ..."
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Cited by 4 (3 self)
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Let r ≥ 3 and (ln n) −1/(r−1) ≤ α ≤ r −3. We show that: Every runiform graph on n vertices with at least αnr ⌊ /r! edges contains a complete rpartite graph with r − 1 parts of size α (ln n) 1/(r−1)⌋ and one part of size n1−αr−2 ⌉. This result follows from a more general digraph version: Let U1
ON SOME EXTREMAL PROBLEMS ON rGRAPHS
, 1971
"... Denote by G(r)(n; k) an rgraph of n vertices and k rtuples. TurLn’s classical problem states: Determine the smallest integer f(n;r, I) so that every G(‘)(n; f (n; r, I)) contains a K(‘)(I). Turin determined f (n; r, I) for r = 2, but nothing is known for r> 2. Put lim,=., f (n; r, l)/(y) = cr ..."
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Cited by 29 (0 self)
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Denote by G(r)(n; k) an rgraph of n vertices and k rtuples. TurLn’s classical problem states: Determine the smallest integer f(n;r, I) so that every G(‘)(n; f (n; r, I)) contains a K(‘)(I). Turin determined f (n; r, I) for r = 2, but nothing is known for r> 2. Put lim,=., f (n; r, l
ON INTEGRAL COMPLETE R−PARTITE GRAPHS
"... AgraphG is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper we investigate integral complete r−partite graphs Kp1,p2,...,pr = Ka1p1,a2p2,...,asps with s ≤ 4. New sufficient conditions for complete 3partite graphs and complete 4partite graphs to be integral ..."
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AgraphG is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper we investigate integral complete r−partite graphs Kp1,p2,...,pr = Ka1p1,a2p2,...,asps with s ≤ 4. New sufficient conditions for complete 3partite graphs and complete 4partite graphs
Integral Complete rpartite Graphs
 DISCRETE MATH
, 2004
"... A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete rpartite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that th ..."
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Cited by 2 (0 self)
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A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete rpartite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved
HDecompositions of rgraphs when H is an rgraph with exactly 2 edges
, 2010
"... Given two rgraphs G and H, an Hdecomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. The minimum number of parts in an Hdecomposition of G is denoted by φr H (G). By a 2edgedecomposition of an rgraph we mean an Hde ..."
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Given two rgraphs G and H, an Hdecomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. The minimum number of parts in an Hdecomposition of G is denoted by φr H (G). By a 2edgedecomposition of an rgraph we mean an H
Orientable Step Domination of Complete rPartite Graphs
"... This paper provides lower orientable kstep domination number and upper orientable kstep domination number of complete rpartite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete rpartite graphs. ..."
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This paper provides lower orientable kstep domination number and upper orientable kstep domination number of complete rpartite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete rpartite graphs.
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 778 (5 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NP
Results 1  10
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610,657