Discrete Mathematics最新文献

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Characterizing forbidden pairs for the existence of even factors

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-03 DOI: 10.1016/j.disc.2024.114384

Lei Zhang , Liming Xiong

Let

H

be a class of given graphs. We say that a graph is

H

f r e e

if it contains no induced subgraph isomorphic to H for every

H \in H

. In 2017, the second author characterized all connected graphs H with order at least three such that every H-free graph G has an even factor if and only if

δ (G) \geq 2

and every odd branch-bond of G has an edge branch. In this paper, we consider the case H is disconnected and characterize all the pairs

{R, S}

of connected graphs such that every

{R, S}

-free graph G has an even factor if and only if

δ (G) \geq 2

and every odd branch-bond of G has an edge branch.

引用次数: 0

Some bounds on the Laplacian eigenvalues of token graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-03 DOI: 10.1016/j.disc.2024.114382

C. Dalfó , M.A. Fiol , A. Messegué

The k-token graph

F_{k} (G)

of a graph G on n vertices is the graph whose vertices are the

(\begin{matrix} n \\ k \end{matrix})

k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G.

It is known that the algebraic connectivity (or second Laplacian eigenvalue) of

F_{k} (G)

equals the algebraic connectivity

α (G)

of G.

In this paper, we give some bounds on the (Laplacian) eigenvalues of the k-token graph (including the algebraic connectivity) in terms of the h-token graph, with

h \leq k

. For instance, we prove that if λ is an eigenvalue of

F_{k} (G)

, but not of G, then

λ \geq k α (G) - k + 1 .

As a consequence, we conclude that if

α (G) \geq k

, then

α (F_{h} (G)) = α (G)

for every

h \leq k

引用次数: 0

Linked partition ideals and overpartitions

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-31 DOI: 10.1016/j.disc.2024.114380

Nancy S.S. Gu, Kuo Yu

Linked partition ideals which were first introduced by Andrews have recently appeared in a series of works to study generating functions for partitions. Recently, Andrews found some relations between a certain kind of overpartitions and 4-regular partitions into distinct parts. Then with the aid of linked partition ideals for overpartitions, Andrews and Chern established a general relation between these two sets of partitions. Motivated by their work, we consider the overpatitions denoted by

A^{k}

satisfying the following conditions: (1) Only odd parts may be overlined; (2) The difference between any two parts is

⩾ 2 k

where the inequality is strict if the larger one is overlined. Let S be a set of given parts. Then

A_{S}^{k}

denotes the subset of overpartitions in

A^{k}

where parts from S are forbidden. Combining linked partition ideals and a recurrence relation for a family of multiple series given by Chern, we study the generating functions for

A_{S}^{k}

for some given S. Furthermore, by establishing a q-series identity, we find a relation between

A_{{\overline{1}}}^{1}

and distinct partitions. Meanwhile, some statistics on partitions are discussed.

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引用次数: 0

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

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