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An improved upper bound on the covering radius of the logarithmic lattice of Q(ζn) Q（ζn）对数格覆盖半径的改进上界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-01 DOI: 10.1016/j.disc.2025.114909

James Punch

Let

R^{m}

be endowed with the Euclidean metric. The covering radius of a lattice

Λ \subset R^{m}

is the least distance r such that, given any point of

R^{m}

, the distance from that point to Λ is not more than r. Lattices can occur via the unit group of the ring of integers in an algebraic number field

K

, by applying a logarithmic embedding

K^{⁎} \to R^{m}

. In this paper, we examine those lattices which arise from the cyclotomic number field

Q (ζ_{n})

, for a given positive integer

n \geq 5

such that

n ≢ 2 (\mod 4)

. We then provide improvements to a result of de Araujo in [3], and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2], and show that, asymptotically, it can be improved no further.

让Rm被赋予欧几里德度规。晶格Λ∧Rm的覆盖半径是最小距离r，使得给定Rm的任意一点，从该点到Λ的距离不大于r。通过应用对数嵌入K→Rm，可以通过代数数域K中的整数环的单位群出现晶格。在本文中，对于给定正整数n≥5，我们研究了由分环数域Q（ζn）产生的格，使得n 2（mod4）。然后，我们对[3]中de Araujo的结果进行了改进，并以n及其不同素数因子的个数给出了该格的覆盖半径的上界。特别地，我们改进了[3，引理2]，并证明，渐近地，它不能再改进了。

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引用次数: 0

Binary trees with extremal number of maximal independent sets 具有极大独立集的极数的二叉树

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-02 DOI: 10.1016/j.disc.2025.114911

Opeyemi Oyewumi , Adriana Roux , Stephan Wagner

A binary tree (more precisely, an unrooted binary tree) is a tree in which all internal vertices (i.e., non-leaves) are exactly of degree 3. We give an upper bound and a lower bound for the number of maximal independent sets in binary trees together with a characterization of the extremal binary trees. The binary trees with second largest number of maximal independent sets are also characterized.

二叉树（更准确地说，是一棵无根二叉树）是一棵所有内部顶点（即非叶子）都恰好是3度的树。给出了二叉树中最大独立集个数的上界和下界，并给出了极值二叉树的特征。对最大独立集个数第二多的二叉树也进行了刻画。

引用次数: 0

Flag-transitive 2-(v,k,λ) designs with λ ≥ (r,λ)2 λ ≥ （r,λ）2的Flag-transitive 2-(v,k,λ)设计

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-29 DOI: 10.1016/j.disc.2025.114949

Junchi Zhang , Jianbing Lu , Meizi Ou

This paper is devoted to the study of 2-designs with

λ \geq {(r, λ)}^{2}

admitting a flag-transitive automorphism group G. The group G has been shown to be point-primitive of either almost simple or affine type. In this paper, we classify the 2-designs with

λ \geq {(r, λ)}^{2} > 1

admitting a flag-transitive almost simple automorphism group with socle

{PSL}_{n} (q)

{PSU}_{n} (q)

for

n \geq 3

本文研究了λ≥(r,λ)2的2-设计，其中包含一个标志传递自同构群G，证明了群G是几乎简单或仿射型的点基元。本文对λ≥(r,λ)2>；1的2-设计进行了分类，这些2-设计承认一个对于n≥3具有集合PSLn(q)或PSUn(q)的flag-传递几乎简单自同构群。

引用次数: 0

A sufficient condition for complete multipartite graphs to be of Type 1 完备多部图为类型1的充分条件

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-11-24 DOI: 10.1016/j.disc.2025.114896

Aseem Dalal, Bhawani Sankar Panda

<div><div>The total chromatic number, <math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math>, of a graph G is the minimum number of colors required to totally color G. A graph G is of Type 1 is <math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math> and of Type 2 if <math><msup><mrow><mi>χ</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>2</mn></math>. A 1992 result of Chew and Yap shows that all complete multipartite graphs of odd order are of Type 1. For graphs of even order, a long-standing conjecture by Hoffman and Rodger states that a complete multipartite graph G of even order is of Type 2 if and only if either it is regular bipartite, or its deficiency, <math><mi>d</mi><mi>e</mi><mi>f</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo></math>, is less than the number of parts of odd size. For a complete multipartite graph <math><mi>G</mi><mo>=</mo><mi>G</mi><mo>[</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>]</mo></math> of even order (where <math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>…</mo><mo>=</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>|</mo><mo>=</mo><mi>r</mi><mo><</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>|</mo><mo>≤</mo><mo>…</mo><mo>≤</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>|</mo></math>), the conjecture has been verified in several cases: when <math><mi>m</mi><mo>≤</mo><mn>4</mn></math>, when <math><mi>p</mi><mo>≤</mo><mn>6</mn></math> or, when <math><mi>d</mi><mi>e</mi><mi>f</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> is very large. In this paper, we prove that the conjecture holds for all such graphs G satisfying<math><mfrac><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo><</mo><mrow><mo>{</mo><mtable><mtr><mtd><mn>1</mn><mo>/</mo><mn>2</mn></mtd><mtd><mtext>when </mtext><mi>r</mi><mtext> is even, </mtext></mtd></mtr><mtr><mtd><mn>1</mn><mo>/</mo><mn>3</mn></mtd><mtd><mtext>when </mtext><mi>r</mi><mtext> is odd, </mtext></mtd></mtr></mtable></mrow></math> thereb

图G的总色数χ″(G)是使G完全上色所需的最小颜色数。图G属于类型1的是χ″(G)=Δ(G)+1，类型2的是χ″(G)=Δ(G)+2。1992年Chew和Yap的结果表明，所有奇阶完全多部图都是类型1。对于偶阶图，Hoffman和Rodger的一个长期猜想指出，偶阶的完全多部图G是类型2当且仅当它是正则二部图，或者它的缺陷def(G)=∑v∈v (G)(Δ（G)−dG(v)）小于奇数大小的部分数。对于偶阶完全多部图G=G[V1,V2，…，Vp]（其中|V1|=…=|Vm|=r<|Vm+1|≤…≤|Vp|），在m≤4、p≤6或def(G)非常大的情况下验证了该猜想。在本文中，我们证明了当r为偶数时G满足p<；{1/2，当r为奇数时G满足1/3，该猜想成立，从而确定了它们的类型。当p≥4时，我们的结果恢复了Yap和Chew的定理，当p≥7时，我们的结果暗示了Rodger和作者的结果。

引用次数: 0

A bridge between Desargues' and Pappus' theorems 是连接德斯格尔定理和帕普斯定理的桥梁

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-11-24 DOI: 10.1016/j.disc.2025.114889

Ákos G.Horváth

In this paper, we investigate the configuration theorems of Desargues and Pappus in a synthetic geometric way. We provide a bridge between the two configurations with a third one that can be considered a specification for both. We do not use the theory of collineations or the analytic description of the plane over a ternary ring.

本文用综合几何方法研究了Desargues和Pappus的组态定理。我们在这两种配置之间提供了一个桥梁，第三种配置可以被认为是两种配置的规范。我们不使用共线理论或三元环上平面的解析描述。

引用次数: 0

A new conjecture on the inertia of graphs 关于图的惯性的一个新猜想

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-22 DOI: 10.1016/j.disc.2025.114953

Saieed Akbari , Clive Elphick , Hitesh Kumar , Shivaramakrishna Pragada , Quanyu Tang

Let G be a graph with adjacency matrix

A (G)

. We conjecture that

2 n^{+} (G) \leq n^{-} (G) (n^{-} (G) + 1),

where

n^{+} (G)

and

n^{-} (G)

denote the number of positive and negative eigenvalues of

A (G)

, respectively. This conjecture generalizes to all graphs the well-known absolute bound for strongly regular graphs. The conjecture also relates to a question posed by Torgašev. We prove the conjecture for special graph families, including line graphs and planar graphs, and provide examples where the conjecture is exact. We also conjecture that for any connected graph G, its line graph

L (G)

satisfies

n^{+} (L (G)) \leq n^{-} (L (G)) + 1

, and obtain partial results.

设G为邻接矩阵a (G)的图。我们推测2n+(G)≤n−(G)(n−(G)+1)，其中n+(G)和n−(G)分别表示A(G)的正特征值和负特征值的个数。这个猜想将众所周知的强正则图的绝对界推广到所有图。这个猜想也与Torgašev提出的一个问题有关。我们证明了特殊图族的猜想，包括线形图和平面图，并给出了猜想是精确的例子。我们还推测，对于任意连通图G，其线形图L(G)满足n+(L(G))≤n−(L(G))+1，并得到部分结果。

引用次数: 0

On forbidding graphs as traces of hypergraphs 关于禁止图作为超图的轨迹

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-01 DOI: 10.1016/j.disc.2025.114907

Dániel Gerbner , Michael E. Picollelli

We say that a hypergraph

H

contains a graph H as a trace if there exists some set

S \subset V (H)

such that

H |_{S} = {h \cap S : h \in E (H)}

contains a subhypergraph isomorphic to H. We study the largest number of hyperedges in 3-uniform hypergraphs avoiding some graph F as trace. In particular, we improve a bound given by Luo and Spiro in the case

F = C_{4}

, and obtain exact bounds for large n when F is a book graph.

如果存在某个集合S∧V(H)使得H bb 0 S={H∩S: H∈E(H)}包含与H同构的子超图H，则我们说超图H包含图H作为迹。我们研究3-一致超图中避免图F作为迹的最大超边数。特别地，我们改进了Luo和Spiro在F=C4情况下给出的界，得到了当F为book图时n大时的精确界。

引用次数: 0

Minimizing beyond-birthday-bound secure permutation-based pseudorandom function 最小化超越生日界限的安全基于置换的伪随机函数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-11-28 DOI: 10.1016/j.disc.2025.114903

Ping Zhang , Honggang Hu , Peng Wang , Yiyuan Luo , Lifa Wu

How can one optimise existing permutation-based pseudorandom functions (PRFs) to minimize the number of components (such as keys, permutations, and inverse permutations) while ensuring beyond-birthday-bound (BBB) security has always been an open problem. This paper focuses on this open problem and proposes a minimal BBB secure permutation-based PRF. Based on PDMMAC, we utilize a single permutation just with forward calls instead of forward and backward calls to construct an improved single-keyed permutation-based PRF, called PDM+. To present better security analyses, we generalize the traditional sum-capture lemma to more general settings. Finally, we prove that PDM+ ensure BBB security up to about

2^{2 n / 3}

adversarial construction and primitive queries by the expectation method and generalized sum-capture lemmas.

如何优化现有的基于排列的伪随机函数（prf）以最小化组件（如密钥、排列和逆排列）的数量，同时确保超出生日界限（BBB）的安全性一直是一个悬而未决的问题。本文针对这一开放问题，提出了一种基于最小BBB安全排列的PRF。在PDMMAC的基础上，我们利用单键前向调用而不是前向和后向调用来构建改进的基于单键排列的PRF，称为PDM+。为了提供更好的安全性分析，我们将传统的和捕获引理推广到更一般的设置。最后，我们通过期望方法和广义和捕获引理证明了PDM+保证BBB的安全性高达22n/3左右的对抗构造和原始查询。

引用次数: 0

A new class of self-orthogonal linear codes and their applications 一类新的自正交线性码及其应用

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-08 DOI: 10.1016/j.disc.2025.114935

Yaozong Zhang, Dabin Zheng, Xiaoqiang Wang

Self-orthogonal codes are a subclass of linear codes that are contained within their dual codes. Since self-orthogonal codes are widely used in quantum codes, lattice theory and linear complementary dual (LCD) codes, they have received continuous attention and research. In this paper, we construct a class of self-orthogonal codes by using the defining-set approach, and determine their weight distributions and the parameters of their dual codes. Some considered codes are optimal according to the tables of best codes known maintained at [11] and a class of almost maximum distance separable (AMDS) codes from their duals is obtained. As applications, we obtain a class of new quantum codes, which are MDS or AMDS according to the quantum Singleton bound under certain conditions. Some examples show that the constructed quantum codes have the better parameters than known ones maintained at [6]. Furthermore, a new class of LCD codes is given, which are almost optimal according to the sphere packing bound.

自正交码是包含在对偶码内的线性码的一个子类。由于自正交码在量子码、晶格理论和线性互补对偶（LCD）码中得到了广泛的应用，一直受到人们的关注和研究。本文利用定义集方法构造了一类自正交码，并确定了它们的权值分布及其对偶码的参数。根据已知最优码表在[11]处保持的最佳码，得到了一类与其对偶的几乎最大距离可分离码（AMDS）。作为应用，我们在一定条件下根据量子单态界得到了一类新的量子码，即MDS或AMDS。一些实例表明，所构建的量子码在[6]下具有比已知量子码更好的参数。在此基础上，给出了一类新的LCD编码，它几乎是最优的。

引用次数: 0

Partitioning planar graphs without 4-cycles and 6-cycles into two disjoint subcubic forests 将没有4环和6环的平面图划分为两个不相交的亚立方森林

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-29 DOI: 10.1016/j.disc.2025.114955

Ziwen Huang , Jian-Bo Lv

Given two positive integers

k_{1}

and

k_{2}

, an

(F_{k_{1}}, F_{k_{2}})

-partition of a graph G is that the set of vertices of G is partitioned into two disjoint subsets

V_{1}, V_{2}

such that for each

i \in {1, 2}

, the induced subgraph

G [V_{i}]

is a forest of maximum degree at most

k_{i}

. Let

G_{4, 6}

be the set of planar graphs with neither 4-cycles nor 6-cycles, and for a positive integer k, let

Δ_{k}

denote a graph of maximum degree at most k. If a graph

G \in G_{4, 6}

, it was proved by Nakprsit, Sittitrai and Pimpasalee that G admits a

(Δ_{3}, Δ_{4})

-partition [[12], Discrete Appl. Math., 356 (2024) 44-51]. In this paper, we prove that every graph in

G_{4, 6}

admits an

(F_{3}, F_{3})

-partition which strengthens the result of Nakprsit, Sittitrai and Pimpasalee [12].

给定两个正整数k1和k2，图G的一个（Fk1,Fk2）划分是将G的顶点集划分为两个不相交的子集V1，V2，使得对于每一个i∈{1,2}，诱导子图G[Vi]是一个最多ki的最大度的森林。设G4,6为既非4圈又非6圈的平面图集合，对于正整数k，设Δk表示最大度不超过k的图。若图G∈G4,6，则由Nakprsit， Sittitrai和Pimpasalee证明G允许（Δ3，Δ4）-划分[[12]，Discrete appll]。数学。科学通报，356(2024):44-51。在本文中，我们证明了G4,6中的每个图都存在一个(F3,F3)-分割，这加强了Nakprsit， Sittitrai和Pimpasalee[12]的结果。

引用次数: 0

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