燃烧数的界
组合数学
2016-11-03 v2
摘要
受旨在衡量图中传染蔓延速度的图论过程启发，Bonato、Janssen 和 Roshanbin [Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22] 将图 G G G b ( G ) b(G) b ( G ) k k k x 1 , … , x k x_1,\ldots,x_k x 1 , … , x k G G G u u u i ∈ { 1 , … , k } i\in \{ 1,\ldots,k\} i ∈ { 1 , … , k } d i s t G ( u , x i ) ≤ k − i {\rm dist}_G(u,x_i)\leq k-i dist G ( u , x i ) ≤ k − i i , j ∈ { 1 , … , k } i,j\in \{ 1,\ldots,k\} i , j ∈ { 1 , … , k } d i s t G ( x i , x j ) ≥ j − i {\rm dist}_G(x_i,x_j)\geq j-i dist G ( x i , x j ) ≥ j − i n n n G G G b ( G ) ≤ 2 ⌈ n ⌉ − 1 b(G)\leq 2\left\lceil\sqrt{n}\right\rceil-1 b ( G ) ≤ 2 ⌈ n ⌉ − 1 b ( G ) ≤ ⌈ n ⌉ b(G)\leq \left\lceil\sqrt{n}\right\rceil b ( G ) ≤ ⌈ n ⌉ n n n G G G 0 < ϵ < 1 0<\epsilon<1 0 < ϵ < 1 b ( G ) ≤ 32 19 ⋅ n 1 − ϵ + 27 19 ϵ b(G)\leq \sqrt{\frac{32}{19}\cdot \frac{n}{1-\epsilon}}+\sqrt{\frac{27}{19\epsilon}} b ( G ) ≤ 19 32 ⋅ 1 − ϵ n + 19 ϵ 27 b ( G ) ≤ 12 n 7 + 3 ≈ 1.309 n + 3 b(G)\leq \sqrt{\frac{12n}{7}}+3\approx 1.309 \sqrt{n}+3 b ( G ) ≤ 7 12 n + 3 ≈ 1.309 n + 3 n n n n 2 n_2 n 2 n ≥ 3 n_{\geq 3} n ≥ 3 T T T b ( T ) ≤ ⌈ ( n + n 2 ) + 1 4 + 1 2 ⌉ b(T)\leq \left\lceil\sqrt{(n+n_2)+\frac{1}{4}}+\frac{1}{2}\right\rceil b ( T ) ≤ ⌈ ( n + n 2 ) + 4 1 + 2 1 ⌉ b ( T ) ≤ ⌈ n ⌉ + n ≥ 3 b(T)\leq \left\lceil\sqrt{n}\right\rceil+n_{\geq 3} b ( T ) ≤ ⌈ n ⌉ + n ≥ 3 r + 1 r+1 r + 1 r r r
引用
@article{arxiv.1511.06023,
title = {Bounds on the Burning Number},
author = {Stéphane Bessy and Anthony Bonato and Jeannette Janssen and Dieter Rautenbach},
journal= {arXiv preprint arXiv:1511.06023},
year = {2016}
}