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1.
We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd
s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ( Kr,s, n) such that every n-term graphic sequence π=( d1, d2,…, dn) with term sum σ(π)= d1+ d2++ dnσ( Kr,s, n) is potentially Kr,s-graphic, where Kr,s is a r× s complete bipartite graph, i.e., π has a realization G containing Kr,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially Kr,s-graphic, and then we determine σ( Kr,r, n) for r=3,4. 相似文献
2.
Suppose G is a graph. The chromatic Ramsey number rc( G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let Mn = min[ rc( G): χ( G) = n]. It was conjectured by Burr et al. (1976) that Mn = ( n − 1) 2 + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M6 and M7. 相似文献
3.
Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph ( S, E) with vertex set S and edge set E such that for u, vS, uvE if and only if u+ vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ( G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ( Kn− E( Kr)) for r2 n/3−1, (ii) obtain a lower bound for ζ( Kn− E( Kr)) when 2 r<2 n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ( Kr,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of Kr,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211). 相似文献
4.
The Ramsey number r( H, Kn) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r( K2,m, Kn)( m−1+o(1))( n/log n) 2 and r( C2m, Kn) c( n/log n) m/(m−1) for m fixed and n→∞. Also r( K2,n, Kn)=Θ( n3/log 2 n) and
. 相似文献
5.
A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of Kn such that E( G) ∩ E(φ( G))=. It is known that all trees T with n (≥2) vertices and T K1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of Kn such that E( G) ∩ E(φ( G))={ e};and that it is 2-embeddable if,for every pair e1, e2 of edges, there is an automorphism φ of Kn such that E( G) ∩ E(φ( G))={ e1, e2}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K1,n−1 are 2-embeddable. In a certain sense, this result is sharp. 相似文献
6.
The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V1, V2,…, Va such that for each i at most d vertices in V1 … Vi have neighbors in Vi+1 and r( Kk, Vi) p | V( G) |, where Vi denotes the subgraph of G induced by Vi. Then there exists a number c depending only on p, d, and k such that r( Kk, G) c | V( G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V( G) such that each component of G− I has at most d vertices and at most two neighbors in I. Then r( G, G) c | V( G) |, where c is a number depending only on d. As a special case, r( G, G) 6 | V( G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4. 相似文献
7.
An ( r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain Kn in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph KN has a coloring which is not ( r, n)-split is denoted by ƒ r( n). Balanced ( r, n)-colorings are defined as edge r-colorings of KN such that every subset of [ N/ r] vertices contains a monochromatic Kn in all colors. Then gr( n) is defined as the smallest N such that KN has a balanced ( r, n)-coloring. The definitions imply that fr( n) gr( n). The paper gives estimates and exact values of these functions for various choices of parameters. 相似文献
8.
A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x0,…, x so that, for each ordinal β < , there exists a strictly increasing finite sequence ( ij) 0 j n of ordinals such that i0 = β, in = and xij+1 is adjacent with xij and with all neighbors of xij in the subgraph of G induced by { xy: β γ }. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets. 相似文献
9.
A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h( n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h( n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs: - (1) Pn, a path on n vertices;
- (2) K1,ns, the graph obtained from K1,n by subdividing an edge once;
- (3) K2,ne, the graph obtained from K2,n by deleting an edge;
- (4) K2,n+, the graph obtained from K2,n by adding an edge between the two degree-n vertices x1 and x2, and a pendent edge at each xi.
Two applications of this result are also discussed in the paper. 相似文献
10.
If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=( a1, a2,…, ak), in which ai denotes the number of edges of color i for i=1,2,…, k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph Kn,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of Kn,n into multicolored trees. 相似文献
11.
Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1, 1999, pp. 387–400) considered a variation of the classical Turán-type extremal problems as follows: For a given graph H, determine the smallest even integer σ( H, n) such that every n-term graphic sequence π=( d1, d2,…, dn) with term sum σ( π)= d1+ d2++ dnσ( H, n) has a realization G containing H as a subgraph. In this paper, for given integers k and ℓ, ℓ7 and 3 kℓ, we completely determine the smallest even integer σ( kCℓ, n) such that each n-term graphic sequence π=( d1, d2,…, dn) with term sum σ( π)= d1+ d2++ dnσ( kCℓ, n) has a realization G containing a cycle of length r for each r, krℓ. 相似文献
12.
图G的最大匹配的路变换图NM(G)是这样一个图,它以G的最大匹配为顶点,如果两个最大匹配M_1与M_2的对称差导出的图是一条路(长度没有限制),那么M_1和M_2在NM(G)中相邻.研究了这个变换图的连通性,分别得到了这个变换图是一个完全图或一棵树或一个圈的充要条件. 相似文献
13.
We consider the following model Hr( n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = ( r − 1) n. Each r-subset of V × ( r−1U) is chosen to be an edge of H ε Hr( n, p) with probability p = p( n), all choices being independent. It is shown that for every 0 < < 1 if P = ( C ln n)/ nr−1 with C = C() sufficiently large, then almost surely every subset V1 V of size | V1 | = (1 − ) n is matchable, that is, there exists a matching M in H such that every vertex of V1 is contained in some edge of M. 相似文献
15.
A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover G's edges. G's bipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d( G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d 2, and investigate the forbidden graphs for d n that have the fewest vertices. We note that for complete graphs, d( Kn) = [log 2n], η( Kn) = d( Kn) for n 16, and η( Kn) is unbounded. The list of minimal forbidden induced subgraphs for η 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices. 相似文献
16.
We generalize two results of Hilton on total-colourings of a graph. The first generalized result unifies several previous results/proof techniques of Bermond, Chen, Chew, Fu, Hilton, Wang, and Yap. Applying the second generalized result, we prove that if G Kn, n is such that Δ( G) = n − 1 and the complement of G with respect to Kn, n contains a 1-factor, then its total chromatic number is Δ( G) + 1. 相似文献
17.
A random graph Gn( x) is constructed on independent random points U1,…, Un distributed uniformly on [0,1] d, d1, in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0< x<1. The connectivity distance cn, the smallest x for which Gn( x) is connected, is shown to satisfy For d2, the random graph Gn( x) behaves like a d-dimensional version of the random graphs of Erdös and Rényi, despite the fact that its edges are not independent: cn/ dn→1, a.s., as n→∞, where dn is the largest nearest-neighbor link, the smallest x for which Gn( x) has no isolated vertices. 相似文献
18.
Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a1g1+ a2g2+ · · · + angn, β = b1h1+ b2h2+ · · · + bmhm ∈ R[ M] satisfy αβ ∈ nil( R[M]), then aibj ∈ nil( R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil( R[ M]) = nil( R)[ M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[ M] in case R is M-π-Armendariz. 相似文献
19.
A new method is developed for finite element (FE) domain decomposition. This method employs a hybrid graph-genetic algorithm for graph partitioning and correspondingly bisects finite element (FE) meshes. A weighted incidence graph is first constructed for the FE mesh, denoted by G0. A coarsening process is then performed using heavy-edge matching. A sequence of such operations is employed in “n” steps, which leads to the formation of Gn with a size suitable for genetic algorithm applications. Hereafter, Gn is bisected using conventional genetic algorithm. The shortest route tree algorithm is used for the formation of the initial population in genetic algorithm. Then an uncoarsening process is performed and the results are transferred to the graph Gn−1. The initial population for genetic algorithm on Gn−1is constructed from the results of Gn. This process is repeated until G0 is obtained in the uncoarsening operation. Employing the properties of G1, the graph G0 is bisected by the genetic algorithm. 相似文献
20.
A 1-factor M of a cubic graph G is strong if | M∩ T|=1 for each 3-edge-cut T of G. It is proved in this paper that a cubic graph G has precisely three strong 1-factors if and only if the graph can be obtained from K4 via a series of ↔ Y operations. Consequently, the graph G admits a Hamilton weight and is uniquely edge-3-colorable. 相似文献
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