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Some of the results of the X - ring ( Road )

Author: WuQiong
Tutor: SunZhiRen
School: Nanjing Normal University
Course: Operational Research and Cybernetics
Keywords: Three connected graph X- ring X- Road X- longest ring X - longest road X- control ring X- Control Road X- can circle graph Degree and κ- decomposition σ_κ (X)
CLC: O157.5
Type: Master's thesis
Year: 2007
Downloads: 26
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Let G = (V, E) n points in three connected graph, so that X (?) V (G). C for G in the ring, for G arbitrary ring C 'have | X ∩ V (C) | ≥ | X ∩ V (C') |, the ring C is called the longest X-ring. We use α (G) the independence number of a graph G, α (X) represents the independence number of G [X]. When k> α (X) seasonal σ k (X) when k ≤ α (X) can form independent set of k points degrees (in G) minimum value of X; σ k (X) = k (n-α (X)). Let X (?) V (G), P is the way in Figure G, if X (?) V (P), then P is called the X-channel. If n is a positive integer and N = the Σ i = 1 K n i and the the N i ≥ 2 ( 1 ≤ i ≤ k), wherein n i (1 ≤ i ≤ k) are integers, called (n to 1 , n 2 , ..., n k ) as a n, k-decomposition. [4] Zou Park made the following conclusions: G n points connected graph, C is the longest cycles in G R = G \\ C, if σ 4 (G) ≥ n 6, then G [R] in any of a communicating branch H satisfy | V (H) | ≤ 2. Conditions in the σ 4 (G) to make changes that take V (G), a subset of X, σ 4 (G) improved as σ < sub> 4 (X), while on the conclusions for further discussion we get the first conclusion: G for three connected graph of the n points, and X (?) V (G), C is an X- The longest cycle, if σ 4 (X) ≥ n, then for any G \\ C a connected component of H have | V (H) ∩ X | ≤ 2, and with exactly two The point X in the communicating branch is not a point outside the X-containing. The easy Seeing conclusion of [4] Zou Park conclusions promotion. [8] Enomoto et al: G n points connected graph σ 3 (G) ≥ n or α (G) ≤ 2, then G Hamilton Road or G in the longest cycles are strong control circle. This conclusion part of the improvement, obtained second conclusion of this paper: G is a connected graph of n points, X (?) V (G), if α (X) ≤ 2, then G with X-road. Visible second conclusion of this paper to promote some of the conclusions of Enomoto Theorem. Conclusion: G n points connected graph is given in [6] Chen Yaojun, if σ 3 (G) ≥ (3n-5) / 2, then G Hamilton Road. Their improvement obtained herein third conclusion: G is a connected graph of the n points, X (?) V (G), if the the σ 3 (X) ≥ (3n-5) / 2, then G X-Road. This conclusion conclusions promotion Chenyao Jun et al. Similarly, Robert Jahansson given in [7]: G n points connected graph, given n a k-decomposition (n 1 n 2 ... , n k ) and n i (1 ≤ i ≤ k) is odd, if G path P so that (?) v ∈ V (G) \\ V (P) are not successive adjacent points on P and d P (v) ≥ [n 1 / 2] [n disjoint paths the n k point. Improvement This conclusion was this fourth conclusion: G n points connected graph X (?) V (G) | X | = s (s 1 , s 2 , ..., s k ) s, a k-decomposition s i (1 ≤ i ≤ k) is odd. Let G be a path P such that any x ∈ X \\ V (P) has | N P (x) ∩ X | ≥ [s to 1 / 2] [s 2 / 2] ... [s k / 2] and x in P will not have one after another neighbor point in X, then G bit cross road P < sub> 1 , P 2 , ..., P k so | the P i ∩ X | the = s i < / sub> (i = 1,2 ..., k). Easy see Robert Jahansson conclusions for fourth conclusion inference. In the last chapter, we propose some future research may be thinking about the problem.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Combinatorics ( combinatorics ) > Graph Theory
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