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1.
The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T, thus (T,T) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.  相似文献
2.
The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we improve Shi’s upper bound for the Laplacian spectral radius of irregular graphs and present some new bounds for the Laplacian spectral radius of some classes of graphs.  相似文献
3.
For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ sk and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.  相似文献
4.
Plesnik in 1972 proved that an (m − 1)-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m − 1 edges. Alder et al. in 1999 showed that if G is a regular (2n + 1)-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and (or) excluding a set of edges, where 1 ≤ kn/2. In particular, we generalize Plesnik’s result and the results obtained by Liu et al. in 1998, and improve Katerinis’ result obtained 1993. Furthermore, we show that the results in this paper are the best possible. The work is partially supported by FRG, Hong Kong Baptist University NSFC (60673047) and SRFDP (20040422004) of China  相似文献
5.
The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.  相似文献
6.
A graph G is dot-critical if contracting any edge decreases the domination number. Nader Jafari Rad (2009) [3] posed the problem: Is it true that a connected k-dot-critical graph G with G=0? is 2-connected? In this note, we give a family of 1-connected 2k-dot-critical graph with G=0? and show that this problem has a negative answer.  相似文献
7.
 In this paper we study three-color Ramsey numbers. Let K i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G 1, G 2 and G 3, if r(G 1, G 2)≥s(G 3), then r(G 1, G 2, G 3)≥(r(G 1, G 2)−1)(χ(G 3)−1)+s(G 3), where s(G 3) is the chromatic surplus of G 3; (ii) (k+m−2)(n−1)+1≤r(K 1,k , K 1,m , K n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed mk≥2, there is a constant c such that r(K k,m , K k,m , K n )≤c(n/logn), and r(C 2m , C 2m , K n )≤c(n/logn) m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55  相似文献
8.
Let G = (V,E) be a connected simple graph. A labeling f: V → ℤ2 induces an edge labeling f*: E → ℤ2 defined by f*(xy) = f(x)+ f(y) for each xyE. For i ∈ ℤ2, let υ f (i) = |f −1(i)| and e f (i) = |f*−1(i)|. A labeling f is called friendly if |υ f (1) − υ f (0)| ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by i f (G) = e f (1) − e f (0). The set {i f (G) | f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI(G). In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.  相似文献
9.
The notion of super-edge-graceful graphs was introduced by Mitchem and Simoson in 1994.However, few examples except trees are known. In this paper, we exhibit two classes of infinitely many cubic graphs which are super-edge-graceful. A conjecture is proposed.  相似文献
10.
The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively.  相似文献
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