共查询到20条相似文献,搜索用时 109 毫秒
1.
A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that \({\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}}{n - 1} \\2\end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}}n \\2\end{array}} \right) - 1\), and \({\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}}{n - 2} \\2\end{array}} \right) + 2\). Next, we investigate the total rainbow connection numbers of graphs G with |V(G)| = n, diam(G) ≥ 2, and clique number ω(G) = n ? s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem. 相似文献
2.
Mohammad Zarrin 《Archiv der Mathematik》2011,96(3):225-226
For any group G, let C(G){\mathcal{C}(G)} denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn{\mathcal{C}_n}-group) if |C(G)| = n{|\mathcal{C}(G)| = n}. In this note, we show that the derived length of a soluble Cn{\mathcal{C}_n}-group (not necessarily finite) is bounded by a function of n. 相似文献
3.
Hong Lin Xiao-feng Guo 《应用数学学报(英文版)》2007,23(1):155-160
Let φ(G),κ(G),α(G),χ(G),cl(G),diam(G)denote the number of perfect matchings,connectivity,independence number,chromatic number,clique number and diameter of a graph G,respectively.In this note,by constructing some extremal graphs,the following extremal problems are solved:1.max{φ(G):|V(G)|=2n,κ(G)≤k}=k[(2n-3)!!],2.max{φ(G):|V(G)|=2n,α(G)≥k}=[multiply from i=0 to k-1(2n-k-i)[(2n-2k-1)!!],3.max{φ(G):|V(G)|=2n,χ(G)≤k}=φ(T_(k,2n))T_(k,2n)is the Turán graph,that is a complete k-partite graphon 2n vertices in which all parts are as equal in size as possible,4.max{φ(G):|V(G)|=2n,cl(G)=2}=n1,5.max{φ(G):|V(G)|=2n,diam(G)≥2}=(2n-2)(2n-3)[(2n-5)!!],max{φ(G):|V(G)|=2n,diam(G)≥3}=(n-1)~2[(2n-5)!!]. 相似文献
4.
Michael Elkin 《Israel Journal of Mathematics》2011,184(1):93-128
The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction
with |S| = W(nlog32)|S| = \Omega ({n^{{{\log }_3}2}}) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound
of Behrend is
|S| = W( [(n)/(22?2 ?{log2n} ·log1/4n)] ).|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }} \cdot {{\log }^{1/4}}n}}} \right). 相似文献
5.
Let
G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let
W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to
[`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A
p
(G); p > 0; we denote a class of functions f analytic in G and satisfying the condition
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