Discrete Mathematics最新文献

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On simultaneous (s,s + t,s + 2t,…)-core partitions 在并发（s,s + t,s + 2t，…）-core分区上

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2025-12-22 DOI: 10.1016/j.disc.2025.114958

William J. Keith , Rishi Nath , James A. Sellers

We consider simultaneous

(s, s + t, s + 2 t, \dots, s + p t)

-core partitions in the large-p limit, or (when

s < t

), partitions in which no hook may be of length

s (\mod t)

. We study generating functions, containment properties, and congruences when s is not coprime to t. As a boundary case of the general study made by Cho, Huh and Sohn, we provide enumerations when s is coprime to t, and answer positively a conjecture of Fayers on the polynomial behavior of the size of the set of simultaneous

(s, s + t, s + 2 t, \dots, s + p t)

-core partitions when p grows arbitrarily large. Of particular interest throughout is the comparison to the behavior of simultaneous

(s, t)

-cores.

我们考虑在大p极限下并发的（s,s+t,s+2t，…，s+pt）-核分区，或者（当s<；t）分区中没有钩子长度为s（modt）的分区。我们研究了s不与t互素时的生成函数、包含性质和同余。作为Cho， Huh和Sohn一般研究的一个边界情况，我们提供了s与t互素时的枚举，并肯定地回答了Fayers关于同时（s,s+t,s+2t，…，s+pt）核分区集在p任意大时的多项式行为的猜想。特别有趣的是与同时（s,t）核的行为进行比较。

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引用次数: 0

On cyclic P(4n,2n − 1)-designs 循环P(4n,2n − 1)-设计

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-16 DOI: 10.1016/j.disc.2026.115000

Wannasiri Wannasit

We show that the generalized Petersen graphs

P (4 n, 2 n - 1)

admits a

ρ^{+}

-labeling for every positive integer n. In this way, we obtain the existence of a cyclic

P (4 n, 2 n - 1)

-decomposition of

K_{v}

for every

v \equiv 1 (\mod 24 n)

我们证明了广义Petersen图P（4n,2n−1）对于每一个正整数n都有一个ρ+标记。这样，我们得到了对于每一个v≡1(mod24n)， Kv存在一个循环P（4n,2n−1）分解。

引用次数: 0

Proof of a conjecture of Green and Liebeck on codes in symmetric groups 关于对称群中码的Green和Liebeck猜想的证明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-15 DOI: 10.1016/j.disc.2026.114999

Teng Fang, Jinbao Li

<div><div>Let A and B be subsets of a finite group G and r a positive integer. If for every <math><mi>g</mi><mo>∈</mo><mi>G</mi></math>, there are precisely r pairs <math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>∈</mo><mi>A</mi><mo>×</mo><mi>B</mi></math> such that <math><mi>g</mi><mo>=</mo><mi>a</mi><mi>b</mi></math>, then B is called a code in G with respect to A and we write <math><mi>r</mi><mi>G</mi><mo>=</mo><mi>A</mi><mo>⋅</mo><mi>B</mi></math>. If in addition B is a subgroup of G, then we say that B is a subgroup code in G. In this paper we resolve a conjecture by Green and Liebeck [8, Conjecture 2.3] on certain subgroup codes in the symmetric group <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. Let <math><mi>n</mi><mo>></mo><mn>2</mn><mi>k</mi></math> and let j be such that <math><msup><mrow><mn>2</mn></mrow><mrow><mi>j</mi></mrow></msup><mo>⩽</mo><mi>k</mi><mo><</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msup></math>. Suppose that X is a conjugacy class in <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> containing x, and <math><msub><mrow><mi>Y</mi></mrow><mrow><mi>k</mi></mrow></msub></math> is the subgroup <math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>×</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></math> of <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, where the factor <math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub></math> permutes the subset <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></math> and the factor <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></math> permutes the subset <math><mo>{</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math>. We prove that <math><mi>r</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>X</mi><mo>⋅</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>k</mi></mrow></msub></math> for some positive integer r if and only if the cycle type of x has exactly one cycle of length <math><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi></mrow></msup></math> for <math><mn>0</mn><mo>⩽</mo><mi>i</mi><mo>⩽</mo><mi>j</mi></math> and all other cycles have length at least <math><mi>k</mi><mo>+</mo><mn>1</mn></math>. We also propose several problems concerning the existence of certain subgroup codes in a finite group G with respect to a conjugation-closed subset i

设A和B是有限群G的子集，r是正整数。如果对于每一个g∈g，恰好有r对(a,b)∈A×B使得g=ab，那么b就被称为g中关于a的一个码，我们写成rG= a·b。如果另外B是G的子群，则我们说B是G中的子群码。本文解决了Green和Liebeck[8，猜想2.3]关于对称群Sn中某些子群码的一个猜想。设n>；2k和j满足2j≤k<；2j+1。设X是Sn中包含X的共轭类，Yk是Sn的子群Sk×Sn−k，其中因子Sk置换子集{1，…，k}，因子Sn−k置换子集{k+1，…，n}。证明对于正整数r， rSn=X⋅Yk当且仅当X的循环类型恰好有一个长度为2i的循环，且对于0≤i≤j，所有其他循环的长度至少为k+1。对于G中的共轭闭子集，给出了有限群G中某些子群码的存在性问题。

引用次数: 0

Solution to a 3-path isolation problem for subcubic graphs 次三次图的三路径隔离问题的解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-13 DOI: 10.1016/j.disc.2025.114970

Karl Bartolo, Peter Borg, Dayle Scicluna

The 3-path isolation number of a connected n-vertex graph G, denoted by

ι (G, P_{3})

, is the size of a smallest subset D of the vertex set of G such that the closed neighbourhood

N [D]

of D intersects the vertex sets of the 3-vertex paths of G, meaning that no two edges of

G - N [D]

intersect. If G is not a 3-path or a 3-cycle or a 6-cycle, then

ι (G, P_{3}) \leq 2 n / 7

. This was proved by Zhang and Wu, and independently by Borg in a slightly extended form. The bound is attained by infinitely many connected graphs having induced 6-cycles. Huang, Zhang and Jin showed that if G has no 6-cycles, or G has no induced 5-cycles and no induced 6-cycles, then

ι (G, P_{3}) \leq n / 4

unless G is a 3-path or a 3-cycle or a 7-cycle or an 11-cycle. They asked if the bound still holds asymptotically for connected graphs having no induced 6-cycles. Thus, the problem essentially is whether induced 6-cycles solely account for the difference between the two bounds. In this paper, we solve this problem for subcubic graphs, which need to be treated differently from other graphs. We show that if G is subcubic and has no induced 6-cycles, then

ι (G, P_{3}) \leq n / 4

unless G is a copy of one of 12 particular graphs whose orders are 3, 7, 11 and 15. The bound is sharp.

连通的N顶点图G的3路隔离数，用ι(G,P3)表示，是G的顶点集的最小子集D的大小，使得D的闭邻域N[D]与G的3顶点路径的顶点集相交，即G−N[D]没有两条边相交。若G不是3径、3环或6环，则ι(G,P3)≤2n/7。这是由Zhang和Wu证明的，Borg以一个稍微扩展的形式独立地证明了。该界是由无穷多个具有诱导6环的连通图得到的。Huang， Zhang和Jin证明了如果G没有6环，或者G没有诱导5环和诱导6环，那么除非G是3径或3环或7环或11环，否则ι(G,P3)≤n/4。他们问，对于没有诱导6环的连通图，界是否仍然是渐近成立的。因此，问题本质上是诱导的6环是否完全解释了两个界之间的差异。在本文中，我们解决了亚三次图的这个问题，它需要区别于其他图的处理。我们证明了如果G是次立方的并且没有诱导的6环，那么ι(G,P3)≤n/4，除非G是阶数为3,7,11和15的12个特定图之一的副本。边界是尖锐的。

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引用次数: 0

On the size and structure of certain subsequence sum set 某子序列和集的大小和结构

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-05 DOI: 10.1016/j.disc.2025.114975

Jagannath Bhanja

We study the following arithmetic questions regarding subsequence sums: How large is the set of subsequence sums, where each element of this set is a sum of at least s (a fixed number) distinct terms, and what are the sets for which the subsequence sum set minimizes? We prove a near-optimal lower bound for the size of this subsequence sum set over the group of residues modulo an odd prime p. We then establish the optimal lower bound for the size of this subsequence sum set over the group of integers and characterize the optimal sequences that achieve this lower bound.

我们研究了以下关于子序列和的算术问题：子序列和的集合有多大，其中该集合的每个元素是至少s（固定数量）个不同项的和，以及子序列和集合最小的集合是什么？我们证明了这个子序列和集的大小在残数群上模一个奇素数p的近最优下界。然后我们建立了这个子序列和集的大小在整数群上的最优下界，并描述了实现这个下界的最优序列。

引用次数: 0

A power sum expansion for the Kromatic symmetric function 罗曼对称函数的幂和展开式

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2025-12-18 DOI: 10.1016/j.disc.2025.114957

Laura Pierson

<div><div>The chromatic symmetric function <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math> is a symmetric function generalization of the chromatic polynomial of a graph, introduced by Stanley [7]. Stanley [7] gave an expansion formula for <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math> in terms of the power sum symmetric functions <math><msub><mrow><mi>p</mi></mrow><mrow><mi>λ</mi></mrow></msub></math> using the principle of inclusion-exclusion, and Bernardi and Nadeau [1] gave an alternate p-expansion for <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math> in terms of acyclic orientations. Crew, Pechenik, and Spirkl [3] defined the Kromatic symmetric function <math><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math> as a K-theoretic analogue of <math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math>, constructed in the same way except that each vertex is assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. They defined a K-analogue <math><msub><mrow><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>λ</mi></mrow></msub></math> of the power sum basis and computed the first few coefficients of the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-expansion of <math><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math> for some small graphs G. They conjectured that the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-expansion always has integer coefficients and asked whether there is an explicit formula for these coefficients. In this note, we give a formula for the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-expansion of <math><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math>, show two ways to compute the coefficients recursively (along with examples), and prove that the coefficients are indeed always integers. In a more recent paper [6], we use our formula from this note to give a combinatorial description of the <math><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></math>-coefficients <math><mo>[</mo><msub><mrow><mover><mrow><mi>p</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>λ</mi></mrow></msub><mo>]</mo><msub><mrow><mover><mrow><mi>X</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>G</mi></mrow></msub></math> and a simple characterization of their signs in the case of unweighted graphs.</

色对称函数XG是由Stanley[7]引入的图的色多项式的对称函数推广。Stanley[7]利用包含-排斥原理给出了XG在幂和对称函数pλ下的展开式，Bernardi和Nadeau[1]给出了XG在无环取向下的另一个p展开式。Crew， Pechenik和Spirkl[3]定义了矩阵对称函数X - G作为XG的k理论模拟，以相同的方式构造，除了每个顶点被分配一个非空的颜色集，以便相邻的顶点具有不重叠的颜色集。他们定义了一个幂和基的k -模拟p - λ，并计算了一些小图形G的X - G的p -展开的前几个系数。他们推测p -展开总是有整数系数，并问这些系数是否有一个显式公式。在这篇文章中，我们给出了X - G的p -展开的一个公式，展示了两种递归计算系数的方法（以及例子），并且证明了系数确实总是整数。在最近的一篇论文[6]中，我们使用本笔记中的公式给出了p -系数[p - λ]X - G的组合描述，并在无加权图的情况下给出了它们的符号的简单表征。

引用次数: 0

An improved bound for equitable proper labellings 公平正确标签的改进界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2025-12-18 DOI: 10.1016/j.disc.2025.114956

Julien Bensmail , Clara Marcille

For every graph G with size m and no connected component isomorphic to

K_{2}

, we prove that, for

L = (1, 1, 2, 2, \dots, ⌊ m / 2 ⌋ + 2, ⌊ m / 2 ⌋ + 2)

, we can assign labels of L to the edges of G in an injective way so that no two adjacent vertices of G are incident to the same sum of labels. This implies that every such graph with size m can be labelled in an equitable and proper way with labels from

{1, \dots, ⌊ m / 2 ⌋ + 2}

, which improves on a result proved by Haslegrave, and Szabo Lyngsie and Zhong, implying this can be achieved with labels from

{1, \dots, m}

对于每一个大小为m且无连通分量同构于K2的图G，我们证明，对于L=(1,1,2,2，…，⌊m/2⌋+2，⌊m/2⌋+2)，我们可以将L的标记以内射的方式分配给G的边，使得G的两个相邻顶点不隶属于相同的标记和。这意味着每一个大小为m的图都可以用{1，…，⌊m/2⌋+2}的标签来合理地标记，这改进了Haslegrave、Szabo Lyngsie和Zhong证明的结果，表明这可以用{1，…，m}的标签来实现。

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引用次数: 0

A note on 2-quasi-bisection of cubic graphs with oddness 2 奇数为2的三次图的2-拟对分的一个注记

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2025-12-08 DOI: 10.1016/j.disc.2025.114939

Siyan Liu , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang

A k-bisection (resp. k-quasi-bisection) of a bridgeless cubic graph G is a vertex 2-coloring satisfying: (i) the sizes of the two color classes are equal (resp. the sizes of the two color classes differ by at most 2), and (ii) the order of each connected component induced by each color class is at most k. Esperet et al. (2017) [7] conjectured that any cubic graph admits a 2-quasi-bisection. In this paper, we prove this conjecture for all bridgeless cubic graphs with oddness 2.

k-二分法。无桥三次图G的k-拟对分（k-拟对分）是顶点2着色，满足：(i)两个颜色类的大小相等(相对于；两个颜色类的大小相差不超过2)，并且（ii）每个颜色类诱导的每个连通分量的阶数不超过k。Esperet et al.(2017)[7]推测任何三次图都允许2-拟对分。本文对奇异数为2的所有无桥三次图证明了这一猜想。

引用次数: 0

Strictly k-colorable graphs 严格k色图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-09 DOI: 10.1016/j.disc.2026.114980

Evan Leonard

Zhu [5] introduced a refined scale of choosability in 2020 and observed that the four color theorem is tight on this scale. We formalize and explore this idea of tightness in what we call strictly colorable graphs. We then characterize all strictly colorable complete multipartite graphs.

Zhu[5]在2020年引入了一个精细的可选择性尺度，并观察到四色定理在这个尺度上是紧密的。我们在所谓的严格可色图中形式化并探索紧性的概念。然后我们刻画了所有严格可着色完全多部图。

引用次数: 0

Spectral extrema of graphs with fixed size: Forbidden star forests 固定尺寸图的谱极值：禁星森林

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-05-01 Epub Date: 2026-01-06 DOI: 10.1016/j.disc.2025.114976

Yanting Zhang , Ligong Wang

The spectral radius of a graph G, denoted by

ρ (G)

, is the largest eigenvalue of its adjacency matrix. The Brualdi-Hoffman-Turán type problem is to determine the maximum spectral radius among all m-edge graphs which do not contain specific forbidden subgraphs. Denote by

S_{ℓ}

the star on

ℓ + 1

vertices. Let F be a star forest, where

F = \cup_{i = 1}^{k} S_{ℓ_{i}}

with

k \geq 2

and

ℓ_{i} \geq 1

for

i \in [k]

. In this paper, we study the Brualdi-Hoffman-Turán type problem for star forests, and prove that if G is an F-free graph with size m, then its spectral radius satisfies

ρ (G) \leq \frac{1}{2} (k - 2 + \sqrt{4 m - k^{2} + 2 k})

, with equality if and only if

G = K_{k - 1} \lor (\frac{m}{k - 1} - \frac{k - 2}{2}) K_{1}

, provided that

m \geq {(2 k - 1)}^{2} {(\sum_{i = 1}^{k} ℓ_{i} + k - 2)}^{2}

图G的谱半径，用ρ(G)表示，是其邻接矩阵的最大特征值。Brualdi-Hoffman-Turán类型问题是确定所有不包含特定禁止子图的m边图的最大谱半径。用S表示在1个顶点上的星号。设F是一个星林，其中对于i∈[k]， F=∪i= 1ks_i，且k≥2且_i≥1。本文研究了星林的Brualdi-Hoffman-Turán型问题，证明了若G是大小为m的无f图，则其谱半径满足ρ(G)≤12(k−2+4m−k2+2k)，且当且仅当G=Kk−1∨(mk−1−k−22)K1，且m≥(2k−1)2(∑i=1k∑i +k−2)2。

引用次数: 0

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