Discrete Mathematics最新文献

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A note on the random triadic process

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-20 DOI: 10.1016/j.disc.2024.114374

Fang Tian , Yiting Yang

<div><div>For a fixed integer <math><mi>r</mi><mo>⩾</mo><mn>3</mn></math>, let <math><msub><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math> be a random r-uniform hypergraph on the vertex set <math><mo>[</mo><mi>n</mi><mo>]</mo></math>, where each r-set is an edge randomly and independently with probability p. The random r-generalized triadic process starts with a complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mi>r</mi><mo>+</mo><mn>2</mn></mrow></msub></math> on the same vertex set, chooses two distinct vertices x and y uniformly at random and iteratively adds <math><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>}</mo></math> as an edge if there is a subset Z with size <math><mi>r</mi><mo>−</mo><mn>2</mn></math>, denoted as <math><mi>Z</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>}</mo></math>, such that <math><mo>{</mo><mi>x</mi><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></math> and <math><mo>{</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></math> for <math><mn>1</mn><mo>⩽</mo><mi>i</mi><mo>⩽</mo><mi>r</mi><mo>−</mo><mn>2</mn></math> are already edges in the graph and <math><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>}</mo></math> is an edge in <math><msub><mrow><mi>H</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math>. The random triadic process is an abbreviation for the random 3-generalized triadic process. Korándi et al. proved a sharp threshold probability for the propagation of the random triadic process, that is, if <math><mi>p</mi><mo>=</mo><mi>c</mi><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math> for some positive constant c, with high probability, the triadic process reaches the complete graph when <math><mi>c</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> and stops at <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math> edges when <math><mi>c</mi><mo><</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math>. In this note, we consider the

引用次数: 0

Larger nearly orthogonal sets over finite fields

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-19 DOI: 10.1016/j.disc.2024.114373

Ishay Haviv , Sam Mattheus , Aleksa Milojević , Yuval Wigderson

For a field

F

and integers d and k, a set

A \subseteq F^{d}

is called k-nearly orthogonal if its members are non-self-orthogonal and every

k + 1

vectors of

A

include an orthogonal pair. We prove that for every prime p there exists some

δ = δ (p) > 0

, such that for every field

F

of characteristic p and for all integers

k \geq 2

and

d \geq k

, there exists a k-nearly orthogonal set of at least

d^{δ \cdot k / \log k}

vectors of

F^{d}

. The size of the set is optimal up to the

\log k

term in the exponent. We further prove two extensions of this result. In the first, we provide a large set

A

of non-self-orthogonal vectors of

F^{d}

such that for every two subsets of

A

of size

k + 1

each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every

k + 1

vectors of the produced set

A

include

ℓ + 1

pairwise orthogonal vectors for an arbitrary fixed integer

1 \leq ℓ \leq k

. The proofs involve probabilistic and spectral arguments and the hypergraph container method.

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引用次数: 0

The number of perfect matchings in a brick

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-18 DOI: 10.1016/j.disc.2024.114365

Fuliang Lu, Huali Pan

A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovász and Plummer.

Lucchesi and Murty conjectured that there exists a positive integer N such that for every

n \geq N

, every brick on n vertices has at least

n - 1

perfect matchings. We present an infinite family of bricks such that for each even integer n (

n > 17

), there exists a brick with n vertices in this family that contains at most

⌈ 0.625 n ⌉

perfect matchings, showing that this conjecture fails.

引用次数: 0

Binary [n,(n ± 1)/2] cyclic codes with good minimum distances from sequences

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-18 DOI: 10.1016/j.disc.2024.114369

Xianhong Xie , Yaxin Zhao , Zhonghua Sun , Xiaobo Zhou

Recently, binary cyclic codes with parameters

[n, (n \pm 1) / 2, \geq \sqrt{n}]

have been a hot topic since their minimum distances have a square-root bound. In this paper, we construct four classes of binary cyclic codes

C_{S, 0}

C_{S, 1}

and

C_{D, 0}

C_{D, 1}

by using two families of sequences, and obtain some codes with parameters

[n, (n \pm 1) / 2, \geq \sqrt{n}]

. For

m \equiv 2 (\mod 4)

, the code

C_{S, 0}

has parameters

[2^{m} - 1, 2^{m - 1}, \geq 2^{\frac{m}{2}} + 2]

, and the code

C_{D, 0}

has parameters

[2^{m} - 1, 2^{m - 1}, \geq 2^{\frac{m}{2}} + 2]

h = 1

and

[2^{m} - 1, 2^{m - 1}, \geq 2^{\frac{m}{2}}]

h = 2

引用次数: 0

Properly colored C4→'s in arc-colored complete and complete bipartite digraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114367

Mengyu Duan , Binlong Li , Shenggui Zhang

A subdigraph of an arc-colored digraph is called properly colored if its every consecutive arcs have distinct colors. Let D be a digraph. For a digraph H, let

p c (D, H)

be the minimum number such that every arc-colored digraph

D^{C}

with

c (D) \geq p c (D, H)

contains a properly colored copy of H, where

c (D)

is the number of colors of

D^{C}

. Let

\overset{\leftrightarrow}{K_{n}}

and

\overset{\leftrightarrow}{K_{m, n}}

be the digraphs obtained from the complete graph

K_{n}

and the complete bipartite graph

K_{m, n}

respectively by replacing each edge uv with a pair of symmetric arcs

(u, v)

and

(v, u)

; and let

\vec{C_{k}}

be the directed cycle of length k. In this paper we determine

p c (\overset{\leftrightarrow}{K_{n}}, \vec{C_{4}})

p c (\overset{\leftrightarrow}{K_{m, n}}, \vec{C_{4}})

and characterize the corresponding extremal arc-colorings of digraphs.

引用次数: 0

On 1-2-3 Conjecture-like problems in 2-edge-coloured graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114368

Julien Bensmail , Hervé Hocquard , Clara Marcille , Sven Meyer

The well-known 1-2-3 Conjecture asks whether almost all graphs can have their edges labelled with

1, 2, 3

so that any two adjacent vertices are distinguished w.r.t. the sums of their incident labels. This conjecture has attracted increasing attention over the last years, with many of its aspects of interest being investigated by several authors. In early 2023, Keusch proposed a full solution to the 1-2-3 Conjecture.

Among other aspects of interest, several works introduced and studied ways of generalising such distinguishing labellings and the 1-2-3 Conjecture to structures more general than graphs, such as digraphs and hypergraphs. In the current work, we introduce two new variants for 2-edge-coloured graphs (having negative and positive edges), in which, through labellings, pairs of adjacent vertices are considered distinguished if and only if the differences between their incident positive and negative sums are different. The difference between the two variants we introduce is that, in one of them, this distinction must be met even when considering the absolute value of these differences.

We investigate how these two variants connect, and how they relate to the original problem. For each of the two variants, we also establish upper bounds on the minimum number of consecutive labels that suffice to design a distinguishing labelling of almost any 2-edge-coloured graph. This leads us to raise some conjectures on this minimum, which, as support, we prove for some families of 2-edge-coloured graphs. We also investigate weaker versions of these conjectures, where one can choose the polarity of the edges.

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引用次数: 0

The intersection density of non-quasiprimitive groups of degree 3p

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114364

Roghayeh Maleki , Andriaherimanana Sarobidy Razafimahatratra

The intersection density of a finite transitive group

G \leq Sym (Ω)

is the rational number

ρ (G)

given by the ratio between the maximum size of a subset of G in which any two permutations agree on some elements of Ω and the order of a point stabilizer of G. In 2022, Meagher asked whether

ρ (G) \in {1, \frac{3}{2}, 3}

for any transitive group G of degree 3p, where

p \geq 5

is an odd prime. If

G \leq Sym (Ω)

is transitive such that

| Ω | = 3 p

, then it is known that

ρ (G) = 1

whenever (a) G is primitive or (b) G is imprimitive and admits a block of size p or at least two G-invariant partitions of Ω. In order to answer Meagher's question, it is left to analyze the intersection density of groups G admitting a unique G-invariant partition

B

whose blocks are of size 3. If G is such a group and

\overline{G}

is the group induced by the action of G on

B

, then we denote the kernel of the canonical epimorphism

G \to \overline{G}

\ker (G \to \overline{G})

. The subgroup

\ker (G \to \overline{G})

is trivial if and only if G is quasiprimitive.

It is shown in this paper that the answer to Meagher's question is affirmative for non-quasiprimitive groups of degree 3p, unless possibly when

p = q + 1

is a Fermat prime and Ω admits a unique G-invariant partition

B

whose blocks are of size 3 such that the induced action

\overline{G}

is an almost simple group with socle equal to

{PSL}_{2} (q)

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引用次数: 0

A characterization of graphs of radius-r flip-width at most 2

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-17 DOI: 10.1016/j.disc.2024.114366

Yeonsu Chang , Sejin Ko , O-joung Kwon , Myounghwan Lee

The radius-r flip-width of a graph, for

r \in N \cup {\infty}

, is a graph parameter defined in terms of a variant of the cops and robber game, called the flipper game, and it was introduced by Toruńczyk (FOCS 2023). We prove that for every

r \in (N ∖ {1}) \cup {\infty}

, the class of graphs of radius-r flip-width at most 2 is exactly the class of (

C_{5}

, bull, gem, co-gem)-free graphs, which are known as totally decomposable graphs with respect to bi-joins.

引用次数: 0

Construction of new linear codes with good parameters from group rings and skew group rings

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-11 DOI: 10.1016/j.disc.2024.114349

Cong Yu, Shixin Zhu

<div><div>In this paper, we use the left principal ideals of group rings and skew group rings to construct linear codes over small finite fields. We study three class of groups: Semidirect product of two cyclic groups, direct product of a cyclic group and semidirect product of two cyclic groups, wreath product of a cyclic group of order n and a cyclic group of order 2. Using these groups, we can get some generator matrices over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. Then by computer search, we obtain 18 new linear codes with parameters <math><msub><mrow><mo>[</mo><mn>32</mn><mo>,</mo><mn>19</mn><mo>,</mo><mn>8</mn><mo>]</mo></mrow><mrow><mn>3</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>36</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>8</mn><mo>]</mo></mrow><mrow><mn>3</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>39</mn><mo>,</mo><mn>21</mn><mo>,</mo><mn>10</mn><mo>]</mo></mrow><mrow><mn>3</mn></mrow></msub></math>,<math><msub><mrow><mo>[</mo><mn>40</mn><mo>,</mo><mn>26</mn><mo>,</mo><mn>8</mn><mo>]</mo></mrow><mrow><mn>3</mn></mrow></msub></math>,<math><msub><mrow><mo>[</mo><mn>40</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>6</mn><mo>]</mo></mrow><mrow><mn>3</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>55</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>22</mn><mo>]</mo></mrow><mrow><mn>3</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>40</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>24</mn><mo>]</mo></mrow><mrow><mn>4</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>40</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>20</mn><mo>]</mo></mrow><mrow><mn>4</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>40</mn><mo>,</mo><mn>27</mn><mo>,</mo><mn>8</mn><mo>]</mo></mrow><mrow><mn>4</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>42</mn><mo>,</mo><mn>29</mn><mo>,</mo><mn>8</mn><mo>]</mo></mrow><mrow><mn>4</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>42</mn><mo>,</mo><mn>25</mn><mo>,</mo><mn>10</mn><mo>]</mo></mrow><mrow><mn>4</mn></mrow></msub></math>,<math><msub><mrow><mo>[</mo><mn>50</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>24</mn><mo>]</mo></mrow><mrow><mn>4</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>50</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>30</mn><mo>]</mo></mrow><mrow><mn>5</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>39</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>19</mn><mo>]</mo></mrow><mrow><mn>5</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>40</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>26</mn><mo>]</mo></mrow><mrow><mn>7</mn></mrow></msub></math>, <math><msub><mrow><mo>[</mo><mn>18</mn><mo>,</mo><mn>12</mn><mo>,</mo><mn>6</mn><mo>]</mo></mrow><mrow><mn>9</mn></mrow></msub></math>, <math><m

引用次数: 0

Edge DP-coloring of planar graphs without 4-cycles and specific cycles

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-11 DOI: 10.1016/j.disc.2024.114353

Patcharapan Jumnongnit , Kittikorn Nakprasit , Watcharintorn Ruksasakchai , Pongpat Sittitrai

The topic of finding sufficient conditions on graphs for their edge list chromatic number to equal their maximum degree has received significant attention in graph theory. Recently, Bernshteyn and Kostochka proposed a generalization of edge list coloring called edge DP-coloring. This development naturally leads to the investigation of similar conditions for edge DP-coloring. In this paper, we find that a graph G has edge DP-chromatic number equal to its maximum degree if (i) G is a planar graph without 4-cycles and 5-cycles, and the maximum degree of G is at least 6; or (ii) G is a planar graph without 4-, 6-cycles, and adjacent 5-cycles, and the maximum degree of G is at least 5. Our results generalize and strengthen previous results in the literature on edge list coloring.

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全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

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