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## Integer K (integer + k)
Kinds of Integer K
## Selected Abstracts## On the number of interior peak solutions for a singularly perturbed Neumann problem COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2007Fang-Hua LinWe consider the following singularly perturbed Neumann problem: where , = , ,2/,x is the Laplace operator, , > 0 is a constant, , is a bounded, smooth domain in ,N with its unit outward normal ,, and f is superlinear and subcritical. A typical f is f(u) = up where 1 < p < +, when N = 2 and 1 < p < (N + 2)/(N , 2) when N , 3. We show that there exists an ,0 > 0 such that for 0 < , < ,0 and for each integer K bounded by where ,N, ,, f is a constant depending on N, ,, and f only, there exists a solution with K interior peaks. (An explicit formula for ,N, ,, f is also given.) As a consequence, we obtain that for , sufficiently small, there exists at least [,N, ,f/,N (|ln ,|)N] number of solutions. Moreover, for each m , (0, N) there exist solutions with energies in the order of ,N,m. © 2006 Wiley Periodicals, Inc. [source] ## Subgraph-avoiding coloring of graphs JOURNAL OF GRAPH THEORY, Issue 4 2010Jia ShenAbstract Given a "forbidden graph" F and an integer k, an F-avoiding k-coloring of a graph G is a k -coloring of the vertices of G such that no maximal F -free subgraph of G is monochromatic. The F-avoiding chromatic numberacF(G) is the smallest integer k such that G is F -avoiding k -colorable. In this paper, we will give a complete answer to the following question: for which graph F, does there exist a constant C, depending only on F, such that acF(G) , C for any graph G? For those graphs F with unbounded avoiding chromatic number, upper bounds for acF(G) in terms of various invariants of G are also given. Particularly, we prove that , where n is the order of G and F is not Kk or . © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300,310, 2010 [source] ## Multi-coloring the Mycielskian of graphs JOURNAL OF GRAPH THEORY, Issue 4 2010Wensong LinAbstract A k -fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent vertices are disjoint. The kth chromatic number of a graph G, denoted by ,k(G), is the minimum total number of colors used in a k -fold coloring of G. Let µ(G) denote the Mycielskian of G. For any positive integer k, it holds that ,k(G) + 1,,k(µ(G)),,k(G) + k (W. Lin, Disc. Math., 308 (2008), 3565,3573). Although both bounds are attainable, it was proved in (Z. Pan, X. Zhu, Multiple coloring of cone graphs, manuscript, 2006) that if k,2 and ,k(G),3k,2, then the upper bound can be reduced by 1, i.e., ,k(µ(G)),,k(G) + k,1. We conjecture that for any n,3k,1, there is a graph G with ,k(G)=n and ,k(µ(G))=n+ k. This is equivalent to conjecturing that the equality ,k(µ(K(n, k)))=n+k holds for Kneser graphs K(n, k) with n,3k,1. We confirm this conjecture for k=2, 3, or when n is a multiple of k or n,3k2/ln k. Moreover, we determine the values of ,k(µ(C2q+1)) for 1,k,q. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 311,323, 2010 [source] ## Unavoidable parallel minors of 4-connected graphs JOURNAL OF GRAPH THEORY, Issue 4 2009Carolyn ChunA parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k -partition triple fan with a complete graph on the vertices of degree k, the k -spoke double wheel, the k -spoke double wheel with axle, the (2k+1)-rung Möbius zigzag ladder, the (2k)-rung zigzag ladder, or Kk. We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 313-326, 2009 [source] ## Sharp bounds for the number of 3-independent partitions and the chromaticity of bipartite graphs JOURNAL OF GRAPH THEORY, Issue 1 2001F. M. DongAbstract Given a graph G and an integer k,,,1, let ,(G,,k) denote the number of k -independent partitions of G. Let ,,,s(p,q) (resp., ,,2,s(p,q)) denote the family of connected (resp., 2-connected) graphs which are obtained from the complete bipartite graph Kp,q by deleting a set of s edges, where p,,,q,,,2. This paper first gives a sharp upper bound for ,(G,3), where G ,,,,,,s(p,q) and 0,,,s,,,(p,,,1)(q,,,1) (resp., G ,,,,,2,s(p,q) and 0,,,s,,,p,+,q,,,4). These bounds are then used to show that if G ,,,,,,s(p,q) (resp., G ,,,,,2,s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ,,,si(p+i,q,i) where max and si,=,s,,,i(p,q+i) (resp., a subset of ,,2,s(p,q), where either 0,,,s,,,q,,,1, or s,,,2q,,,3 and p,,,q,+,4). By applying these results, we show finally that any 2-connected graph obtained from Kp,q by deleting a set of edges that forms a matching of size at most q,,,1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48,77, 2001 [source] ## Rainbow trees in graphs and generalized connectivity NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2010Gary ChartrandAbstract An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 , k , n. A k -rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S , V(T). The minimum number of colors needed in a k -rainbow coloring of G is the k -rainbow index of G. For every two integers k and n , 3 with 3 , k , n, the k -rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection {T1,T2,,,T,} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) , V(Tj) = S for every pair i,j of distinct integers with 1 , i,j , ,. For an integer k with 2 , k , n, the k -connectivity ,k(G) of G is the greatest positive integer , for which G contains at least , internally disjoint trees connecting S for every set S of k vertices of G. It is shown that ,k(Kn)=n,,k/2, for every pair k,n of integers with 2 , k , n. For a nontrivial connected graph G of order n and for integers k and , with 2 , k , n and 1 , , , ,k(G), the (k,,)-rainbow index rxk,,(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least , internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rxk,,(Kn) are determined for all possible values k and , when n , 6. It is also shown that for , , {1, 2}, rx3,,(Kn) = 3 for all n , 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## Approximating the smallest k -edge connected spanning subgraph by LP-rounding NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2009Harold N. GabowAbstract The smallest k-ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k -edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3/k for undirected graphs with k > 1 odd, 1 + 2/k for undirected graphs with k even, and 1 + 2/k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2/k. On the hardness side we show that for some absolute constant c > 0, for any integer k , 2 (k , 1), a polynomial-time algorithm approximating the smallest k -ECSS on undirected (directed) multigraphs to within ratio 1 + c/k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009 [source] ## Conditional diameter saturated graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2008C. BalbuenaAbstract The conditional diameter D,,(G) of a connected graph G is a measure of the maximum distance between two subsets of vertices satisfying a given property ,, of interest. For any given integer k , 1, a connected graph G is said to be conditional diameter k -saturated if D,,(G) , k and there does not exist any other connected graph G, with order ,V(G,), = ,V(G),, size ,E(G,), > ,E(G),, and conditional diameter D,,(G,) , k. In this article, we obtain such conditional diameter saturated graphs for a number of properties ,,, generalizing the results obtained in (Ore, J Combin Theory 5(1968), 75,81) for the (standard) diameter D(G). © 2008 Wiley Periodicals, Inc. NETWORKS, 2008 [source] ## Mutually independent hamiltonian paths in star networks NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2005Cheng-Kuan LinAbstract Two hamiltonian paths P1 = ,u1, u2,,,un(G), and P2 = ,v1, v2,,,vn(G), of G from u to v are independent if u = u1 = v1, v = vn(G) = un(G), and vi , ui for every 1 < i < n(G). A set of hamiltonian paths, {P1, P2,,,Pk}, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. 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 exists k -mutually independent hamiltonian paths between any two nodes from distinct partite sets. The mutually independent hamiltonian laceability of a bipartite graph G, IHPL(G), is the maximum integer k such that G is k -mutually independent hamiltonian laceable. Let Sn denote the n -dimensional star graph. We prove that IHPL(S2) = 1, IHPL(S3) = 0, and IHPL(Sn) = n, 2 if n , 4. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 110,117 2005 [source] ## On the fault-tolerant diameter and wide diameter of ,-connected graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2005Jian-Hua YinAbstract The fault-tolerant diameter, Dk, and wide diameter, dk, are two important parameters for measuring the reliability and efficiency of interconnection networks. It is well known that for any ,-connected graph G and any integer k, 1 , k , ,, we have Dk , dk. However, what we are interested in is how large the difference between dk and Dk can be. For any 2-connected graph G with diameter d, Flandrin and Li proved that d2 , D2 + 1 if d = 2 and d2 , (d , 1)(D2 , 1) if d , 3. In this article, we further prove that d2 , max{D2 + 1, (d , 1)(D2 , d) + 2} for d , ,(D2 , 1)/2, and d2 , max{D2 + 1,,(D2 , 1)2/4, + 2} for d , ,(D2 , 1)/2, + 1, and we also show that this upper bound can be achieved. Moreover, for any ,(, 3)-connected graph G, we prove that d, , D, + 1 if D, , 1 = 2 and d, , max{D, + 2,,(D,)2/4, + 2} if D, , 1 = 2 and D, , 1 , 3. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 88,94 2005 [source] ## More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP RANDOM STRUCTURES AND ALGORITHMS, Issue 4 2008Lars EngebretsenAbstract Samorodnitsky and Trevisan [STOC 2000, pp. 191,199] proved that there exists, for every positive integer k, a PCP for NP with O(log n) randomness, query complexity 2k + k2, free bit complexity 2k, completeness 1 - ,, and soundness 2 + ,. In this article, we devise a new "outer verifier," based on the layered label cover problem recently introduced by Dinur et al. [STOC 2003, pp. 595,601], and combine it with a new "inner verifier" that uses the query bits more efficiently than earlier verifiers. Our resulting theorem is that there exists, for every integer f , 2, every positive integer t , f(f - 1)/2, and every constant , > 0, a PCP for NP with O(log n) randomness, query complexity f + t, free bit complexity f, completeness 1 - ,, and soundness 2 - t + ,. As a corollary, there exists, for every integer q , 3 and every constant , > 0, a q -query PCP for NP with amortized query complexity 1 + + ,. This improves upon the result of Samorodnitsky and Trevisan with respect to query efficiency, i.e., the relation between soundness and the number of queries. Although the improvement may seem moderate,the construction of Samorodnitsky and Trevisan has amortized query complexity 1 + 2/,we also show in this article that combining our outer verifier with any natural candidate for a corresponding inner verifier gives a PCP that is less query efficient than the one we obtain.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 [source] ## The game chromatic number of random graphs RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2008Tom BohmanGiven a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number ,g(G) is the minimum k for which the first player has a winning strategy. In this study, we analyze the asymptotic behavior of this parameter for a random graph Gn,p. We show that with high probability, the game chromatic number of Gn,p is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 [source] ## On the complexity of the circular chromatic number JOURNAL OF GRAPH THEORY, Issue 3 2004H. HatamiAbstract Circular chromatic number, ,c is a natural generalization of chromatic number. It is known that it is NP -hard to determine whether or not an arbitrary graph G satisfies ,(G)=,c(G). In this paper we prove that this problem is NP -hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k,,,2 and n,,,3, for a given graph G with ,(G),=,n, it is NP -complete to verify if . © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 226,230, 2004 [source] ## Rainbow trees in graphs and generalized connectivity NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2010Gary ChartrandAbstract An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 , k , n. A k -rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S , V(T). The minimum number of colors needed in a k -rainbow coloring of G is the k -rainbow index of G. For every two integers k and n , 3 with 3 , k , n, the k -rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection {T1,T2,,,T,} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) , V(Tj) = S for every pair i,j of distinct integers with 1 , i,j , ,. For an integer k with 2 , k , n, the k -connectivity ,k(G) of G is the greatest positive integer , for which G contains at least , internally disjoint trees connecting S for every set S of k vertices of G. It is shown that ,k(Kn)=n,,k/2, for every pair k,n of integers with 2 , k , n. For a nontrivial connected graph G of order n and for integers k and , with 2 , k , n and 1 , , , ,k(G), the (k,,)-rainbow index rxk,,(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least , internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rxk,,(Kn) are determined for all possible values k and , when n , 6. It is also shown that for , , {1, 2}, rx3,,(Kn) = 3 for all n , 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] ## About smoothness of solutions of the heat equations in closed, smooth space-time domains COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2005Hongjie DongWe consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ,2 = {(x1, x2)} of class C2k. We give sufficient conditions for u to have kth -order continuous derivatives with respect to (x1, x2) in D, for integers k , 2. The equation is supplemented with C2k boundary data, and we assume that f , C2(k,1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc. [source] |