Báo cáo toán học: "The Neighborhood Characteristic Parameter for Graphs" pptx

... − 15 + 12 − 3=0 N char(dodecahedron) = 20 − 60 + 20 = −20 2 Computing the Neighborhood Characteristic The neighborhood characteristic of a graph can also be calculated in terms of the complete bipartite ... overcounting is corrected for in the following theorem (which also shows that, for a bipartite graph G, N char(G) equals twice the ‘bipartite characteristic defined in [5]). T...

Ngày tải lên: 07/08/2014, 07:21

... x 1 ∈ N i for 2 ≤ i ≤ d. But N 1 has non-trivial intersection with each of N 2 , . . . , N d , so we must have x 2 ∈ N i for 2 ≤ i ≤ d. Thus x 2 ∈ d  i=1 N i , ... L C (a) ∪ L C (b). By Lemma 3.2, S induces a clique of order m − 1. For y ∈ S, put P y = N(y) ∩ K. From the above we know that for each y ∈ S we have |P y | = 2. If there exists z ∈  y∈S P y , ... be a coloring of a grap...

Ngày tải lên: 07/08/2014, 15:22

Ngày tải lên: 05/08/2014, 15:21

... points at distance 2 from v) for any vertex v of G.Henceα(G) ≥  v∈G d 1 (v)/[1+d 1 (v)+d 2 (v)] ≥  v∈G d 1 (v)/2α(G)(bylemma 1 and the preceding remark). Therefore α(G) 2 ≥ n ¯ d/2orα(G) ≥ √ n ¯ d2 ... α(G) 2 ≥ n ¯ d/2orα(G) ≥ √ n ¯ d2 as claimed. This improves Denley’s Theorems 1 and 2. It is sharp for the regular complete bipartite graphs K aa . The above results are readily extended...

Ngày tải lên: 07/08/2014, 06:20

... uses both colors on K. Therefore, c is a 2-coloring of H(G). For the skeptics of the proof of the 4-Color Theorem and since we do not want to use this powerful theorem for our application if not ... assume henceforth. Suppose that a vertex v ∈ V (G)\V (C) has two neighbors x i , x j on C which are not consecutive vertices of C.Sincek≤7, there is a color a ∈ L(v) such that a = c(x) for a...

Ngày tải lên: 07/08/2014, 06:20

... induction on k, c k,j = d k,j for the given j and all k, and by induction on j this holds for all j. It now remains only to show that d k,k−1 = C k−1 for all k. For this, observe that C k−1 /  2k− ... 2 n/2 extremal permutations for n even and 2 n−1 extremal permutations for n odd. For n even, there is a simple noninductive description: if the permutation is p 0 , p 1 , ,p n−1...

Ngày tải lên: 07/08/2014, 07:21

... a forest of given size. Lemma 2 (i) An r-uniform hypergraph with at least r!(t − 1) r edges contains a forest with t edges. (ii) Let F be a 3-uniform hypergraph. Then either (a) F contains a forest ... [5] conjectured that for any t there is a constant C so that any r-uniform hypergraph with C r edges contains a sunflower with t edges. We can obtain a bound of this form for forests, indeed...

Ngày tải lên: 07/08/2014, 08:22

... maximum weight bases of M with the weight function X → | X| for X ∈ π. This weighted matroidal approach yields a proof for the Berge-type formula of [5] on the size of a maximum k-piece packing. Indeed, ... one can see that all the above considerations for the k-piece packings hold for the (l, u)-piece packings as well, with the necessary modifications. For illustrating this, we bri...

Ngày tải lên: 07/08/2014, 08:22

... al. [2], gives such a lower bound. It states that for some constant c, cr(G) ≥ cG 3 /|G| 2 for every graph G with G ≥ 4|G|. See [22, 25] for recent improvements. Other general lower bound ... number, which is analogous to Theorem 3 for crossing number (without the dependence on degree). Theorem 4. The following are equivalent for a graph G: 1. mcr(G) ≤ c 1 |G| for some constan...

Ngày tải lên: 07/08/2014, 15:22

... say that C(uv) = ∅ if uv /∈ E(G) for u, v ∈ V (G). For a subgraph H of G, we denote C(H) = {C(e) | e ∈ E(H)} and c(H) = |C(H)|. For a vertex v of G, the color neighborhood CN(v) of v is defined ... Let C N (v) denote the color neighborhood of a vertex v of G. In a previous paper, we showed that if |CN (u) ∪ C N (v)| ≥ s (color neighborhood union condition) for every pair of verti...

Ngày tải lên: 07/08/2014, 15:23