Discrete Mathematics最新文献

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On 2-distance 16-coloring of planar graphs with maximum degree at most five

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-30 DOI: 10.1016/j.disc.2024.114379

Zakir Deniz

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that G is a planar graph with a maximum degree at most 5. We prove that G admits a 2-distance 16-coloring, which improves the result given by Zou et al. (2024) [13].

引用次数: 0

Determinantal decomposition of graphs with a unique perfect matching

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-27 DOI: 10.1016/j.disc.2024.114370

Daniel A. Jaume , Diego G. Martinez , Cristian Panelo

In this work, a structural decomposition of graphs with a unique perfect matching is introduced. The decomposition is given by the barbell subgraphs: even subdivisions of two

K_{3}

graphs joined by an edge such that the unique perfect matching of G induces a perfect matching in the subgraphs. The decomposition breaks a graph G, with a unique perfect matching, into two subgraphs, one of which is a Kőnig-Egerváry graph. Furthermore, the decomposition is shown to be multiplicative with respect to determinantal-type (Schur) functions of the adjacency matrix of graphs with a unique perfect matching. Additionally, in this work, Godsil's formula for the determinant of trees with perfect matching is extended to all graphs with a unique perfect matching.

引用次数: 0

Further results and questions on S-packing coloring of subcubic graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-27 DOI: 10.1016/j.disc.2024.114376

Maidoun Mortada , Olivier Togni

For a non-decreasing sequence of integers

S = (a_{1}, a_{2}, \dots, a_{k})

, an S-packing coloring of G is a partition of

V (G)

into k subsets

V_{1}, V_{2}, \dots, V_{k}

such that the distance between any two distinct vertices

x, y \in V_{i}

is at least

a_{i} + 1

1 \leq i \leq k

. We consider the S-packing coloring problem on subclasses of subcubic graphs: For

0 \leq i \leq 3

, a subcubic graph G is said to be i-saturated if every vertex of degree 3 is adjacent to at most i vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and G is said to be

(3, i)

-saturated if every heavy vertex is adjacent to at most i heavy vertices. We prove that every 1-saturated subcubic graph is

(1, 1, 3, 3)

-packing colorable and

(1, 2, 2, 2, 2)

-packing colorable. We also prove that every

(3, 0)

-saturated subcubic graph is

(1, 2, 2, 2, 2, 2)

-packing colorable.

引用次数: 0

On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-27 DOI: 10.1016/j.disc.2024.114378

Eiichi Bannai , Etsuko Bannai , Chin-Yen Lee , Ziqing Xiang , Wei-Hsuan Yu

Finding the maximum cardinality of a 2-distance set in Euclidean space is a classical problem in geometry. Lisoněk in 1997 constructed a maximum 2-distance set in

R^{8}

with 45 points. That 2-distance set constructed by Lisoněk has a distinguished structure of a coherent configuration of type

(2, 2; 3)

and is embedded in two concentric spheres in

R^{8}

. In this paper we study whether there exists any other similar embedding of a coherent configuration of type

(2, 2; 3)

as a 2-distance set in

R^{n}

, without assuming any restriction on the size of the set. We prove that there exists no such example other than that of Lisoněk. The key ideas of our proof are as follows: (i) study the geometry of the embedding of the coherent configuration in Euclidean spaces and to derive diophantine equations coming from this embedding. (ii) solve diophantine equations with certain additional conditions of integrality of some parameters of the combinatorial structure by using the method of auxiliary equations.

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引用次数: 0

A short note on Chen-Qian theorem

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-27 DOI: 10.1016/j.disc.2024.114371

Jiyou Li, Yanghongbo Zhou

We give a shorter proof of Chen-Qian Theorem from a perspective of abstract tube.

引用次数: 0

Existential closure in uniform hypergraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-23 DOI: 10.1016/j.disc.2024.114372

Andrea C. Burgess , Robert D. Luther , David A. Pike

For a positive integer n, a graph with at least n vertices is n-existentially closed or simply n-e.c. if for any set of vertices S of size n and any set

T \subseteq S

, there is a vertex

x \notin S

adjacent to each vertex of T and no vertex of

S ∖ T

. We extend this concept to uniform hypergraphs, find necessary conditions for n-e.c. hypergraphs to exist, and prove that random uniform hypergraphs are asymptotically n-existentially closed. We then provide constructions to generate infinitely many examples of n-e.c. hypergraphs. In particular, these constructions use certain combinatorial designs as ingredients, adding to the ever-growing list of applications of designs.

引用次数: 0

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

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