Discrete Mathematics最新文献

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Brooks-type theorems for relaxations of square colorings 方形着色松弛的布鲁克斯型定理

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114233

Eun-Kyung Cho , Ilkyoo Choi , Hyemin Kwon , Boram Park

The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer h, the proper h-conflict-free chromatic number of a graph G, denoted $χ_{pcf}^{h} (G)$ , is the minimum k such that G has a proper k-coloring where every vertex v has $\min {\deg_{G} (v), h}$ colors appearing exactly once on its neighborhood. Caro, Petruševski, and Škrekovski put forth a Brooks-type conjecture: if G is a graph with $Δ (G) \geq 3$ , then $χ_{pcf}^{1} (G) \leq Δ (G) + 1$ . The best known result regarding the conjecture is $χ_{pcf}^{1} (G) \leq 2 Δ (G) + 1$ , which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all h, and also enlarge the class of graphs for which the conjecture is known to be true.

Our main result is the following: for a graph G, if $Δ (G) \geq h + 2$ , then $χ_{pcf}^{h} (G) \leq (h + 1) Δ (G) - 1$ ; this is tight up to the additive term as we explicitly construct infinitely many graphs G with $χ_{pcf}^{h} (G) = (h + 1) (Δ (G) - 1)$ . We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for h-dynamic coloring.

最近，有人提出了对图的正方形进行适当着色的以下放宽方法：对于正整数 h，图 G 的适当 h-无冲突色度数（表示为 χpcfh(G)）是使 G 具有适当 k 着色的最小 k，在该适当 k 着色中，每个顶点 v 都有 min{degG(v),h} 颜色在其邻域上恰好出现一次。卡罗、佩特鲁舍夫斯基和什克里科夫斯基提出了一个布鲁克斯式猜想：如果 G 是一个Δ(G)≥3 的图，那么 χpcf1(G)≤Δ(G)+1。关于这个猜想的最著名结果是 χpcf1(G)≤2Δ(G)+1，这是由帕赫和塔尔多斯的一个结果暗示的。我们针对所有 h 改进了上述结果，并扩大了已知猜想为真的图类。我们的主要结果如下：对于一个图 G，如果 Δ(G)≥h+2，那么 χpcfh(G)≤(h+1)Δ(G)-1；由于我们明确地构造了具有 χpcfh(G)=(h+1)(Δ(G)-1)的无穷多个图 G，因此这在加法项之前是紧密的。我们还证明了猜想对于弦图是真的，并获得了准线图和无爪图的部分结果。我们的主要结果还改进了布鲁克斯式的 h 动态着色结果。

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引用次数: 0

Induced subgraphs and tree decompositions VI. Graphs with 2-cutsets 诱导子图和树分解 VI.带 2 切集的图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-02 DOI: 10.1016/j.disc.2024.114195

Tara Abrishami , Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl

This paper continues a series of papers investigating the following question: which hereditary graph classes have bounded treewidth? We call a graph t-clean if it does not contain as an induced subgraph the complete graph $K_{t}$ , the complete bipartite graph $K_{t, t}$ , subdivisions of a $(t \times t)$ -wall, and line graphs of subdivisions of a $(t \times t)$ -wall. It is known that graphs with bounded treewidth must be t-clean for some t; however, it is not true that every t-clean graph has bounded treewidth. In this paper, we show that three types of cutsets, namely clique cutsets, 2-cutsets, and 1-joins, interact well with treewidth and with each other, so graphs that are decomposable by these cutsets into basic classes of bounded treewidth have bounded treewidth. We apply this result to two hereditary graph classes, the class of ( $I S K_{4}$ , wheel)-free graphs and the class of graphs with no cycle with a unique chord. These classes were previously studied and decomposition theorems were obtained for both classes. Our main results are that t-clean ( $I S K_{4}$ , wheel)-free graphs have bounded treewidth and that t-clean graphs with no cycle with a unique chord have bounded treewidth.

本文是研究以下问题的系列论文的续篇：哪些遗传图类具有有界树宽？如果一个图的诱导子图不包含完整图 Kt、完整二方图 Kt,t、(t×t)-墙的细分图以及(t×t)-墙的细分图的线图，我们就称该图为 t-洁净图。众所周知，对于某个 t，具有有界树宽（treewidth）的图一定是 t 净图；但是，并不是每个 t 净图都具有有界树宽（treewidth）。在本文中，我们证明了三类切集（即簇切集、2-切集和 1-连接）与树宽以及它们之间的相互作用，因此可由这些切集分解为有界树宽基本类的图都具有有界树宽。我们将这一结果应用于两个遗传图类，即无（ISK4, 轮）图类和无唯一弦循环图类。以前曾对这两类图进行过研究，并得到了这两类图的分解定理。我们的主要结果是：t-clean (ISK4, wheel)-free graphs 具有有界树宽；t-clean graphs with no cycle with a unique chord 具有有界树宽。

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引用次数: 0

Toroidal Hitomezashi patterns 环状人字形图案

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-30 DOI: 10.1016/j.disc.2024.114231

Qiuyu Ren , Shengtong Zhang

Extending a proposal of Defant and Kravitz (2024) [2], we define Hitomezashi patterns and loops on a torus and provide several structural results for such loops. For a given pattern, our main theorems give optimal residual information regarding the Hitomezashi loop length, loop count, as well as possible homology classes of such loops. Special attention is paid to toroidal Hitomezashi patterns that are symmetric with respect to the diagonal $x = y$ , where we establish a novel connection between Hitomezashi and knot theory.

我们扩展了德凡特和克拉维茨（Defant and Kravitz，2024 年）[2] 的提议，定义了环上的 Hitomezashi 图案和环，并提供了此类环的若干结构性结果。对于给定的模式，我们的主要定理给出了有关常陆环长、环数以及此类环可能的同构类的最优残差信息。我们特别关注相对于对角线 x=y 对称的环状人字桥图案，在此我们建立了人字桥与结理论之间的新联系。

引用次数: 0

On the Ramsey number of the double star 关于双星的拉姆齐数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-30 DOI: 10.1016/j.disc.2024.114227

Freddy Flores Dubó, Maya Stein

The double star $S (m_{1}, m_{2})$ is obtained from joining the centres of a star with $m_{1}$ leaves and a star with $m_{2} \leq m_{1}$ leaves. We give a short proof of a new upper bound on the two-colour Ramsey number of $S (m_{1}, m_{2})$ which holds for all $m_{1}, m_{2}$ with $\frac{\sqrt{5} + 1}{2} m_{2} < m_{1} < 3 m_{2}$ . Our result implies that for all positive m, the Ramsey number of the double star $S (2 m, m)$ is at most $⌈ 4.275 m ⌉ + 1$ .

双星 S(m1,m2) 是由一个有 m1 个叶子的星和一个有 m2≤m1 个叶子的星的中心连接而成。我们简短地证明了 S(m1,m2)的双色拉姆齐数的新上界，该上界在所有 m1,m2 为 5+12m2<m1<3m2 时都成立。我们的结果意味着，对于所有正 m，双星 S(2m,m) 的拉姆齐数最多为⌈4.275m↪Pe_2309+1。

引用次数: 0

Asymptotically good LCD 2-quasi-abelian codes over finite fields 有限域上渐近良好的 LCD 2-类阿贝尔码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-29 DOI: 10.1016/j.disc.2024.114224

Guanghui Zhang , Liren Lin , Xuemei Liu

In this paper, we construct a class of linear complementary dual (LCD for short) 2-quasi-abelian codes over a finite field. Based on counting the number of such codes and estimating the number of the codes in this class whose relative minimum weights are small, we prove that the class of LCD 2-quasi-abelian codes over any finite field is asymptotically good. The existence of such codes is unconditional, which is different from the case of self-dual 2-quasi-abelian codes over a special finite field.

在本文中，我们构建了一类有限域上的线性互补对偶（简称 LCD）2-类阿贝尔码。基于对这类编码数量的统计和对该类编码中相对最小权值较小的编码数量的估计，我们证明了任意有限域上的 LCD 2-quasi-abelian 编码类是渐近良好的。这类码的存在是无条件的，这与特殊有限域上的自偶 2- 类阿贝尔码的情况不同。

引用次数: 0

Improved 2-distance coloring of planar graphs with maximum degree 5 最大阶数为 5 的平面图的改进型 2-距离着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-29 DOI: 10.1016/j.disc.2024.114225

Kengo Aoki

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of G is the minimum k such that G has a 2-distance k-coloring, denoted by $χ_{2} (G)$ . In this paper, we show that $χ_{2} (G) \leq 17$ for every planar graph G with maximum degree $Δ (G) \leq 5$ , which improves a former bound $χ_{2} (G) \leq 18$ .

图 G 的 2-distance k-coloring（2-距离 k-着色）是一种适当的 k-着色，使得距离为 2 或更小的任意两个顶点获得不同的颜色。G 的双距色度数是使 G 具有双距 k 着色的最小 k 值，用 χ2(G)表示。本文证明，对于最大度 Δ(G)≤5 的每个平面图 G，χ2(G)≤17，这改进了以前的一个约束 χ2(G)≤18。

引用次数: 0

Upper bounds on the number of colors in interval edge-colorings of graphs 图的区间边着色中颜色数的上界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-28 DOI: 10.1016/j.disc.2024.114229

Arsen Hambardzumyan, Levon Muradyan

An edge-coloring of a graph G with colors $1, \dots, t$ is called an interval t-coloring if all colors are used and the colors of edges incident to each vertex of G are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph G with at least one edge has an interval t-coloring, then $t \leq 2 | V (G) | - 3$ . In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph G admits an interval t-coloring, then $t \leq \frac{11}{6} | V (G) |$ . In the same paper Axenovich suggested a conjecture that if a planar graph G has an interval t-coloring, then $t \leq \frac{3}{2} | V (G) |$ . In this paper we first prove that if a graph G has an interval t-coloring, then $t \leq \frac{| E (G) | + | V (G) | - 1}{2}$ . Next, we confirm Axenovich's conjecture by showing that if a planar graph G admits an interval t-coloring, then $t \leq \frac{3 | V (G) | - 4}{2}$ . We also prove that if an outerplanar graph G has an interval t-coloring, then $t \leq | V (G) | - 1$ . Moreover, all these upper bounds are sharp.

如果使用了所有颜色，并且 G 的每个顶点所带的边的颜色是不同的，并且构成了一个整数区间，那么具有颜色 1,...t 的图 G 的边着色称为区间 t 着色。1990 年，卡马里安证明，如果至少有一条边的图 G 具有区间 t 着色，则 t≤2|V(G)|-3 。2002 年，阿克森诺维奇改进了平面图的这一上限：如果平面图 G 有一个区间 t-着色，那么 t≤116|V(G)| 。在同一篇文章中，阿克森诺维奇提出了一个猜想：如果一个平面图 G 有一个区间 t-着色，那么 t≤32|V(G)|。在本文中，我们首先证明如果一个图 G 有一个区间 t-着色，那么 t≤|E(G)|+|V(G)|-12 。接着，我们证实了阿克森诺维奇的猜想，即如果一个平面图 G 有一个区间 t-着色，那么 t≤3|V(G)|-42.我们还证明，如果外平面图 G 有一个区间 t-着色，那么 t≤|V(G)|-1.此外，所有这些上界都很尖锐。

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引用次数: 0

Subfield codes of CD-codes over F2[x]/〈x3−x〉 F2[x]/〈x3-x〉上的CD编码的子字段编码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-27 DOI: 10.1016/j.disc.2024.114223

Anuj Kumar Bhagat, Ritumoni Sarma, Vidya Sagar

<div>A non-zero <math><mi>F</mi></math>-linear map from a finite-dimensional commutative <math><mi>F</mi></math>-algebra to the field <math><mi>F</mi></math> is called an <math><mi>F</mi></math>-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an <math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-valued trace of the <math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-algebra <math><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>:</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo><mo>/</mo><mo>〈</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mi>x</mi><mo>〉</mo></math> to study binary subfield code <math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup></math> of <math><msub><mrow><mi>C</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><msub><mrow><mo>(</mo><mi>x</mi><mo>⋅</mo><mi>d</mi><mo>)</mo></mrow><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mo>:</mo><mi>x</mi><mo>∈</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>}</mo></math> for each defining set D derived from a certain simplicial complex. For <math><mi>m</mi><mo>∈</mo><mi>N</mi></math> and <math><mi>X</mi><mo>⊆</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></math>, define <math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>v</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>:</mo><mtext>Supp</mtext><mo>(</mo><mi>v</mi><mo>)</mo><mo>⊆</mo><mi>X</mi><mo>}</mo></math> and <math><mi>D</mi><mo>:</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>(</mo><mi>u</mi><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>3</mn></mrow></msub></math>, a subset of <math><msubsup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math>, where <math><mi>u</mi><mo>=</mo><mi>x</mi><mo>+</mo><mo>〈</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mi>x</mi><mo>〉</mo><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mo>{</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>L</mi></mrow><mrow>

从有限维交换 F 代数到 F 域的非零 F 线性映射，如果其内核不包含任何非零理想，则称为 F 值迹。在本文中，我们利用 F2-代数 R2:=F2[x]/〈x3-x〉的 F2 值踪迹来研究 CD:={(x⋅d)d∈D:x∈R2m} 的二进制子域码 CD(2)，对于每个定义集 D 都是从某个单纯复数导出的。对于 m∈N 和 X⊆{1,2,...,m}，定义 ΔX:={v∈F2m:Supp(v)⊆X}和 D:=(1+u2)D1+u2D2+(u+u2)D3，R2m 的一个子集，其中 u=x+〈x3-x〉,D1∈{ΔL,ΔLc},D2∈{ΔM,ΔMc}和 D3∈{ΔN,ΔNc}，对于 L,M,N⊆{1,2,...,m}。这些二进制子字段码在 L、M 和 N 的卡片数的某些温和条件下是最小的。因此，我们得到了一些最小、自正交和距离最优的二进制线性编码无穷族，它们要么是 2 权码，要么是 4 权码。值得一提的是，我们还得到了几种新的距离最优二元线性编码。

引用次数: 0

On extended 1-perfect bitrades 关于扩展的 1-完美比特等级

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-27 DOI: 10.1016/j.disc.2024.114222

Evgeny A. Bespalov, Denis S. Krotov

Extended 1-perfect codes in the Hamming scheme $H (n, q)$ can be equivalently defined as codes that turn to 1-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-4 codes with certain dual distances. We define extended 1-perfect bitrades in $H (n, q)$ in five different manners, corresponding to the different definitions of extended 1-perfect codes, and prove the equivalence of these definitions of extended 1-perfect bitrades. For $q = 2^{m}$ , we prove that such bitrades exist if and only if $n = l q + 2$ . For any q, we prove the nonexistence of extended 1-perfect bitrades if n is odd.

汉明方案 H(n,q) 中的扩展 1-perfect 码可以等价地定义为在任意坐标上穿刺后变为 1-perfect 码的码，具有一定交集数组的完全规则码，具有一定权系数的均匀堆积码，相对于一定反码的直径完美码，具有一定对偶距离的距离-4 码。我们以五种不同的方式定义 H(n,q) 中的扩展 1-perfect bitrades，与扩展 1-perfect 码的不同定义相对应，并证明这些扩展 1-perfect bitrades 定义的等价性。对于 q=2m，我们证明当且仅当 n=lq+2 时存在这样的比特等级。对于任意 q，如果 n 为奇数，我们证明扩展的 1-perfect 比特等级不存在。

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引用次数: 0

Pursuit-evasion in graphs: Zombies, lazy zombies and a survivor 图形中的追逐-逃避：僵尸、懒惰僵尸和幸存者

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-08-26 DOI: 10.1016/j.disc.2024.114220

Prosenjit Bose , Jean-Lou De Carufel , Thomas Shermer

<div>We study zombies and survivor, a variant of the game of cops and robber on graphs where the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The difference is that zombies must follow an edge of a shortest path towards the survivor on their turn. Let <math><mi>z</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> be the smallest number of zombies required to catch the survivor on a graph G with n vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that <math><mi>z</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math>. We also show that there exist maximum-degree-3 outerplanar graphs such that <math><mi>z</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow></math>.A zombie that can remain at its current vertex on its turn is called lazy. Let <math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math> be the smallest number of lazy zombies required to catch the survivor. The ability to remain at its current vertex on its turn makes lazy zombies more powerful than normal zombies but less powerful than cops. We prove that <math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn></math> for connected outerplanar graphs which is tight in the worst case. We also show that in this case, the survivor is caught after <math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math> rounds. We then show that <math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>k</mi></math> for connected graphs with treedepth k and that <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>)</mo></math> rounds are sufficient to catch the survivor. The bound on treedepth implies that <math><msub><mrow><mi>z</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math> is at most <math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math> for connected graphs with treewidth k, <math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math> for connected planar graphs, <math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>g</mi><mi>n</mi></mrow></msqrt><mo>)</mo></math> for connected graphs with genus g and <math><mi>O</mi><mo>(</mo><mi>h</mi><msqrt><mrow><mi>h</mi><mi>n</mi></mrow></msqrt><mo>)</mo></math> for connected graphs with any excluded h-vertex minor. Our results on lazy zombies still hold when an adversary chooses

我们研究的是 "僵尸与幸存者"，这是警察与强盗游戏在图形上的一种变体，其中单个幸存者扮演强盗，并试图从扮演警察的僵尸手中逃脱。所不同的是，僵尸在轮到自己时必须沿着最短路径的一条边走向幸存者。假设 z(G) 是在一个有 n 个顶点的图 G 上抓住幸存者所需的最少僵尸数量。我们证明存在外平面图和简单多边形的可见性图，使得 z(G)=Θ(n) 。我们还证明，存在最大度数为 3 的外平面图，使得 z(G)=Ω(n/log(n))。让 zL(G) 成为捕捉幸存者所需的最小懒惰僵尸数量。由于懒惰僵尸可以停留在当前顶点，所以它比普通僵尸更强大，但比警察更弱小。我们证明，对于连通的外平面图，zL(G)≤2，这在最坏情况下是紧密的。我们还证明，在这种情况下，幸存者会在 O(n) 轮后被抓获。然后我们证明，对于树深度为 k 的连通图，zL(G)≤k，并且 O(n2k) 轮足以捕捉到幸存者。对树深的约束意味着，对于树宽为 k 的连通图，zL(G)最多为 (k+1)logn；对于连通的平面图，zL(G)最多为 O(n)；对于属数为 g 的连通图，zL(G)最多为 O(gn)；对于具有任意排除的 h 个顶点的连通图，zL(G)最多为 O(hhn)。当对手选择僵尸的初始位置时，我们关于懒惰僵尸的结果仍然成立。

引用次数: 0

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