Discrete Mathematics最新文献

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A combinatorial proof of a family of truncated identities for the partition function

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-14 DOI: 10.1016/j.disc.2025.114434

Yongqiang Chen, Olivia X.M. Yao

In 2012, Andrews and Merca proved a truncated partition identity by studying the truncated series of Euler's pentagonal number theorem. Andrews and Merca's work has opened up a new study on truncated theta series and a number of results on truncated theta series have been proved in the past decade. Recently, Xia, Yee and Zhao proved a new truncated partition identity by taking different truncated series than the one chosen by Andrews and Merca. Very recently, Yao proved a new truncated identity on Euler's pentagonal number theorem. The identity is equivalent to a family of truncated identities for the partition function which involves the results proved by Andrew-Merca, and Xia-Yee-Zhao. In this paper, we provide a purely combinatorial proof of the family of truncated identities for the partition function. In particular, we answer a question on combinatorial proofs of two partition identities, which were posed by Wang and Xiao.

引用次数: 0

New MDS codes of non-GRS type and NMDS codes

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-14 DOI: 10.1016/j.disc.2025.114436

Yujie Zhi, Shixin Zhu

Maximum distance separable (MDS) and near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent error-correcting capabilities. This paper focuses on a specific class of linear codes and establishes necessary and sufficient conditions for them to be MDS or NMDS. Additionally, we employ the well-known Schur method to demonstrate that they are non-equivalent to generalized Reed-Solomon codes.

最大距离可分码（MDS）和近最大距离可分码（NMDS）因其代数特性和出色的纠错能力，已被广泛应用于通信系统、数据存储和量子编码等多个领域。本文重点研究了一类特定的线性编码，并建立了它们成为 MDS 或 NMDS 的必要条件和充分条件。此外，我们还利用著名的舒尔方法证明它们与广义里德-所罗门码不等同。

引用次数: 0

Wilf classes for weak ascent sequences avoiding a pair or triple of length-3 patterns

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-14 DOI: 10.1016/j.disc.2025.114438

David Callan , Toufik Mansour

A weak ascent sequence is a word

π = π_{1} π_{2} \dots π_{n}

over the set of nonnegative integers such that

π_{1} = 0

and

π_{i} \leq 1 + weak_asc (π_{1} π_{2} \dots π_{i - 1})

for

i = 2, \dots, n

, where

weak_asc (π_{1} π_{2} \dots π_{m})

is the number of weak ascents in the word

π_{1} π_{2} \dots π_{m}

, that is, the number of two-entry factors

π_{j} π_{j + 1}

such that

π_{j} \leq π_{j + 1}

. Here we obtain some enumerative results for weak ascent sequences avoiding a set of two or three 3-letter patterns, leading to a conjecture for the number of Wilf equivalence classes for weak ascent sequences avoiding a pair (respectively, triple) of 3-letter patterns. The main tool is the use of generating trees. Some cases are treated using bijective methods.

引用次数: 0

A note on the multicolour version of the Erdős-Hajnal conjecture

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-13 DOI: 10.1016/j.disc.2025.114437

Maria Axenovich, Lea Weber

Informally, the multicolour version of the Erdős-Hajnal conjecture (shortly EH-conjecture) asserts that if a sufficiently large host clique on n vertices is edge-coloured avoiding a copy of some fixed edge-coloured clique, then there is a large homogeneous set of size

n^{β}

for some positive β, where a set of vertices is homogeneous if it does not induce all the colours. This conjecture, if true, claims that imposing local conditions on edge-partitions of cliques results in a global structural consequence such as a large homogeneous set, a set avoiding all edges of some part. While this conjecture attracted a lot of attention, it is still open even for two colours.

In this note, we reduce the multicolour version of the EH-conjecture to the case when the number of colours used in a host clique is either the same as in the forbidden pattern or one more. We exhibit a non-monotonicity behaviour of homogeneous sets in coloured cliques with forbidden patterns by showing that allowing an extra colour in the host graph could actually decrease the size of a largest homogeneous set.

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引用次数: 0

Enumerating alternating-runs with sign in Type A and Type B Coxeter groups

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-13 DOI: 10.1016/j.disc.2025.114439

Hiranya Kishore Dey , Sivaramakrishnan Sivasubramanian

<div><div>We enumerate alternating runs in the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⊆</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This leads us to enumerate the bivariate peak and valley polynomial with sign taken into account. We prove an exact formula for this signed enumerator in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and show that this polynomial depends on the value of <span><math><mi>n</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>. If <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the polynomial enumerating alternating runs in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, Wilf showed that <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></math></span> divides <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and determined the exponent of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></math></span> that divides <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. Let <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span> be the polynomials enumerating alternating runs in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> respectively. By finding the exponent of <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>)</mo></math></span> that divides <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, we refine Wilfs result when <span><math><mi>n</mi><mo>≡</mo><mn>2</mn><mo>,</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>. When <span><math><mi>n</mi><mo>≡</mo><mn>0</mn><mo>,</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we show that the exponent is one short of what Wilf obtains.</div><div>As applications of our results, we get moment type identities involving the coefficients of <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>±</mo></mrow></msubsup><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, refinements to enumerating alternating and unimodal permutations in <span><math><msub><mrow><mi>A</

引用次数: 0

On the complement of a signed graph

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-13 DOI: 10.1016/j.disc.2025.114433

Matteo Cavaleri, Alfredo Donno, Stefano Spessato

Given a signed graph

(Γ, σ)

and a spanning tree of Γ, we define pseudo-potentials on Γ, which coincide with usual potential functions in the balanced case. Using a pseudo-potential, we are able to define a signature on the complement

Γ^{c}

of Γ, in such a way that the signed complete graph obtained by taking the union of Γ and

Γ^{c}

is stable under switching equivalence, and providing a solution to an open problem in the literature of signed graphs. Then, we introduce three new notions of signed regularity, we characterize them in terms of the adjacency matrix of

(Γ, σ)

, and we show that under such regularity hypotheses the spectrum of the signed complete graph can be described in terms of the spectra of

(Γ, σ)

and of its signed complement. As an application of our machinery, we define a signed version of a generalization of the classical NEPS of graphs, whose signature is stable under switching equivalence. In particular, this construction allows to give a switching stable definition of the lexicographic product of signed graphs, for which the spectrum is explicitly determined in the regular case.

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引用次数: 0

Spectral bipartite Turán problems on linear hypergraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-13 DOI: 10.1016/j.disc.2025.114435

Chuan-Ming She , Yi-Zheng Fan , Liying Kang

Let F be a graph, and let

B_{r} (F)

be the class of r-uniform Berge-F hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for

B_{r} (C_{3})

-free linear r-uniform hypergraphs and upper bounds for the spectral radii of

B_{r} (K_{2, t})

-free or

{B_{r} (K_{s, t}), B_{r} (C_{3})}

-free linear r-uniform hypergraphs, where

C_{3}

and

K_{s, t}

are respectively the triangle and the complete bipartite graph with one part having s vertices and the other part having t vertices. Our work implies an upper bound for the number of edges of

{B_{r} (K_{s, t}), B_{r} (C_{3})}

-free linear r-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.

引用次数: 0

Properly colored cycles in edge-colored complete graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-12 DOI: 10.1016/j.disc.2025.114403

Tianjiao Dai , Hao Li , Yannis Manoussakis , Qiancheng Ouyang

As an analogy of the well-known anti-Ramsey problem, we study the existence of properly colored cycles of given length in an edge-colored complete graph. Let

pr (K_{n}, G)

be the maximum number of colors in an edge-coloring of

K_{n}

with no properly colored copy of G. In this paper, we determine the exact threshold for cycles

pr (K_{n}, C_{ℓ})

, which proves a conjecture proposed by Fang, Győri, and Xiao, that the maximum number of colors in an edge-coloring of

K_{n}

with no properly colored copy of

C_{ℓ}

\max {(\begin{matrix} ℓ - 1 \\ 2 \end{matrix}) + n - ℓ + 1, ⌊ \frac{ℓ - 1}{3} ⌋ n - (\begin{matrix} ⌊ \frac{ℓ - 1}{3} ⌋ + 1 \\ 2 \end{matrix}) + 1 + r_{ℓ - 1}}

, where

C_{ℓ}

is a cycle on ℓ vertices,

ℓ - 1 \equiv r_{ℓ - 1} \mod 3

, and

0 \leq r_{ℓ - 1} \leq 2

. It is a slight modification of a previous conjecture posed by Manoussakis, Spyratos, Tuza and Voigt. Also, we consider the maximal coloring of

K_{n}

whether a properly colored cycle can be extended by exact one more vertex.

引用次数: 0

On the domination number of graphs with minimum degree at least seven

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-12 DOI: 10.1016/j.disc.2025.114424

Csilla Bujtás , Michael A. Henning

A dominating set in a graph G is a set

S \subseteq V (G)

such that every vertex that is not in S is adjacent to a vertex from it. The domination number

γ (G)

is the minimum cardinality of a dominating set in G. For

k \geq 1

, let

G_{k}

denote the class of all connected graphs with minimum degree at least k. The problem to determine or estimate the best possible constants

c_{dom, k}

(which depend only on k) such that

γ (G) \leq c_{dom, k} \cdot n (G)

for all

G \in G_{k}

is well studied, where

n (G)

denotes the order of G. In this paper we establish best known lower and upper bounds to date on the constant

c_{dom, 7}

, namely

\frac{3}{14} \leq c_{dom, 7} \leq \frac{4}{14}

引用次数: 0

Box complexes: At the crossroad of graph theory and topology

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-12 DOI: 10.1016/j.disc.2025.114422

Hamid Reza Daneshpajouh , Frédéric Meunier

Various simplicial complexes can be associated with a graph. Box complexes form an important family of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their topological properties. They provide thus a fascinating topic mixing topology and discrete mathematics. This paper is intended to provide an up-do-date survey on box complexes. It is based on classical results and recent findings from the literature, but also establishes new results improving our current understanding of the topic, and identifies several challenging open questions.

引用次数: 0

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