Báo cáo toán học: "On Extremal Graphs With No Long Paths" docx

... ✷ Summarizing, we now know all extremal graphs, G with f (G)=2δ +1. The connected extremal graphs are those given in Theorem 3 along with the trivial ones with only 2δ + 1 vertices. We noticed that ... the sense that there exist graphs of minimum degree δ with no longer paths. Our purpose in this article is to characterize all such extremal graphs. For a graph G with...

Ngày tải lên: 07/08/2014, 06:20

... which is not Laplacian integral. Next we consider the case n = 8, and note that without loss of generality, necessarily G must contain the subgraph shown in Figure 24. Note that either none, one, ... there is no graph on 4 vertices with the five Laplacian eigenvalues 0, 1, 2, 3, 4. Hence we must have ∆(G) ≤ 2; as noted in Section 1, there are exactly five Laplacian integral graphs with...

Ngày tải lên: 07/08/2014, 21:20

... it exists); we call it the canonical similarity from C to C ′ . Not every two equivalent graphs a r e isomorphic (e.g. take non- isomorphic strongly regular graphs with the same parameters [3]), ... exist infinitely many non-isomorphic cubic graphs with no separators of cardinality k (see e.g. [10]), we obtain the following result. 2 The adjacency algebra of a Klein scheme belongs...

Ngày tải lên: 08/08/2014, 01:20

Ngày tải lên: 05/08/2014, 09:46

Ngày tải lên: 05/08/2014, 15:20

... Canada. Abstract We prove that for c ≥ 2.522 a random graph with n vertices and m = cn edges is not 3-colorable with probability 1 − o(1). Similar bounds for non-k-colorability are given for k>3. 1991 Mathematics ... to uniformly random pseudographs on a given degree sequence (for a definition see also [15]). In particular, they proved that such a pseudograph with ρn edges is w.h.p. n...

Ngày tải lên: 07/08/2014, 06:20

... rightmost descendant v  is not the start vertex. Then 1. If v ∈ V n is not visited by T , then v  is not visited by T . 2. If v ∈ V n is visited by T only once, then v  is not visited by T . Proof: ... ends in 0. Then let v  be the rightmost descendant of v.Ifv  is not the start vertex, then by Lemma 4.2 v  is not visited. Note that v  is obtained from v by removing a positive number...

Ngày tải lên: 07/08/2014, 06:22

... give a linear f(n, p, q). 1 Introduction 1.1 Notations and definitions For basic graph concepts see the monograph of Bollob´as [1]. V (G)andE(G)denotethe vertex-set and the edge-set of the graph ... vertices. In this paper log n denotes the base 2 logarithm. pr(n) denotes the parity of the natural number n,soitis1ifn is odd and 0 otherwise. 1.2 Edge colorings with at least q colors in every...

Ngày tải lên: 07/08/2014, 07:21

... that cover the vertices of G. It is well known (see, for example, [1]) that for some >0itisimpossibletoap- proximate in polynomial time α(G)and χ(G) within a factor of n  , assuming P = NP. We ... which has many alter- native definitions. One of them is based on the notion of the orthonormal labeling of the graph. Orthonormal labeling of dimension k of the graph G is a mapping f : V (G)...

Ngày tải lên: 07/08/2014, 08:20

... Hoffman-Singleton graphs. The same phenomenon was observed [19] on a graph of order 32, the smallest of the McKay-Miller- ˇ Sir´aˇn graphs for q =2. The remainder of this section deals with non-affine automorphisms ... that F belongs to two Hoffman- Singleton subgraphs, namely to H and also to the subgraph induced by V 0 ∪ V 3 .Tosee that there are no other Hoffman-Singleton subgraphs conta...

Ngày tải lên: 07/08/2014, 08:22