Báo cáo toán học: "Degree powers in graphs with forbidden subgraphs" docx

... , proving (8). Inequality (9) follows easily, since, in fact, for every p>3, the function g (x)hasa local minimum at 0. ✷ 5 H-free graphs In this section we are going to prove the following ... (5), we obtain that (r −1)  n p +1  +  2n p +1  <n. Letting n →∞, we see that p ≥ r, contradicting the assumption and completing the proof. ✷ Maximizing independently each summand in...

Ngày tải lên: 07/08/2014, 08:20

... e G (X) for the number of edges induced by X; - e G (X, Y ) for the number of edges joining vertices in X to vertices in Y ; - G −u for the graph obtained by removing the vertex u ∈ V (G) ; - Γ G (u) ... graph G − u into the sets A = Γ G (u) and B = V (G) \(Γ G (u) ∪ {u}) . Since G contains no C 2k+ 2 , the graph G −u does not contain a path of order 2k + 1 with both ends in A. Applyi...

Ngày tải lên: 08/08/2014, 01:20

... Computing, 8:131– 159, 1999. [10] A. N. Lager˚as, M. Lindholm, A note on the component structure in random inter- section graphs with tunable clustering. The Electronic Journal of Combinatorics, ... 10]. A generalized random intersection graph is introduced in [5] by allowing a more general connection probability in the underlying bipartite graph. The corresponding vertex degrees ar...

Ngày tải lên: 08/08/2014, 11:20

... by analyzing the stabilizers of the possible orbits and providing 2-distinguishing colorings for graphs with 6-orbits, and graphs without 6-orbits, but with 3-orbits. Clearly graphs with largest ... the original, and the original labeling is r-distinguishing if and only if the new graph is rigid. Although a given group might be the automorphism group of graphs with different di...

Ngày tải lên: 07/08/2014, 06:20

... When g = 6 the minimal cubic graph is Heawood’s graph with 14 vertices. (It is the incidence graph of points and lines in the seven point projective plane.) Excising a tree consisting of a vertex ... that constructing Cayley graphs that have triangles, and then collapsing the triangles, can lead to graphs with large girth. Using this ‘collapsing method’ they obtain a graph with 80...

Ngày tải lên: 07/08/2014, 06:22

... minimal digraphs with given girth, Proc. 9th S-E Conf. Combinatorics, Graph Theory and Computing (1978), 181-187. [5] V. Chv´atal and E. Szemer´edi, Short cycles in directed graphs, J. Combin. ... deg + (u)+deg + (v) ≥ 3 for all pairs of distinct vertices u, v in G,theng ≤n/2. 2 Main Results Suppose u is a vertex in G. If a digraph G  is obtained from G by adding to it an extra ve...

Ngày tải lên: 07/08/2014, 07:21

... P 2 , P 3 connecting two vertices a, b, and suppo se that u is in P 1 . The remaining vertices of P 1 (including a and b) can easily be coloured with no bad edges. Colour the internal vertices ... a n−1 we obtain K 5 − e as a minor, which is again a co ntradiction. So supp ose that G does not contain such a path P . Then, applying Claim 2 with H = W n , G has no minor formed as in Cla...

Ngày tải lên: 08/08/2014, 01:20

... we add the centre c to L we do not gain any new edges since every edge is already incident with a leaf but since l(c) > 0 the demand for ingoing edges increases. Hence, L ∪ {c} has negative ... the in nite star. We immediately encounter the problem that the set L of all leaves is incident with in nitely many edges but has in nite demand for ingoing edges, i.e. l(L) = ∞. This results...

Ngày tải lên: 07/08/2014, 21:20

... bound follows by applying the Stirling approximation formula for factorials. It is worth noting that since every -regular graph with density d and 2n vertices contains, in each color class, at ... M. D. Plummer, Matching Theory, Akad´emiai Kiad´o, Budapest, 1986 [6] H. Minc, Nonnegative Matrices, Wiley, 1988 [7] V. R¨odl and A. Ruci´nski, Perfect matchings in -regular graphs and the...

Ngày tải lên: 07/08/2014, 06:22

... distributed as independent Poisson variables with means (d − 1) r /2r,provided d ≤ √ 2logn − 1withg fixed (and independently in [10] for fixed d). In this paper we allow d = d(n)andg = g(n)toincreasewithn, ... combinatorics 11 (2004), #R66 12 Theorem 3 For any d and n such that N(n, d) =0,letJ ⊆ E(K n ). Then, with [x] m denoting the falling factorial and j i the number of edges in J...

Ngày tải lên: 07/08/2014, 08:20