Discrete Mathematics最新文献

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Minimizing the number of edges in (C4,K1,k)-co-critical graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.disc.2025.114409

Gang Chen , Chenchen Ren , Zi-Xia Song

<div><div>Given graphs <math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, a {red, blue}-coloring of the edges of a graph G is a critical coloring if G has neither a red <math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math> nor a blue <math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. A non-complete graph G is <math><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>-co-critical if G admits a critical coloring, but <math><mi>G</mi><mo>+</mo><mi>e</mi></math> has no critical coloring for every edge e in the complement of G. Motivated by a conjecture of Hanson and Toft from 1987, we study the minimum number of edges over all <math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub><mo>)</mo></math>-co-critical graphs on n vertices. We show that for all <math><mi>k</mi><mo>≥</mo><mn>3</mn></math> and <math><mi>n</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mo>⌊</mo><msqrt><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>⌋</mo><mo>+</mo><mn>2</mn></math>, if G is a <math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub><mo>)</mo></math>-co-critical graph on n vertices, then<math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mn>3</mn><mo>−</mo><mfrac><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>k</mi><mo>+</mo><mo>⌊</mo><msqrt><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msqrt><mo>⌋</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>.</mo></math> Moreover, this linear bound is asymptotically best possible for all <math><mi>k</mi><mo>≥</mo><mn>3</mn></math> and <math><mi>n</mi><mo>≥</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>4</mn></math>. It is worth noting that our constructions for the case when k is even have at least three different critical colorings. For <math><mi>k</mi><mo>=</mo><mn>2</mn></math>, we obtain the sharp bound for the minimum number of edges of <math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>)</mo></math>-co-critical graphs on <math><mi>n</mi><mo>≥</mo><mn>5</mn></math> vertices by showing that all such graphs have at le

引用次数: 0

A relation between multiplicity of 1 as a Laplacian eigenvalue and induced matching numbers in trees

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.disc.2025.114401

Zhiwen Wang, Qian-Qian Chen, Ji-Ming Guo, Xiao-Meng Li

The induced matching number of a graph is the largest number of edges at pairwise distance at least 2. Let T be a tree of order n with induced matching number

β^{'} (T)

. In this paper, we give a sharp upper bound in terms of n and

β^{'} (T)

for the multiplicity of 1 as a Laplacian eigenvalue of T. Moreover, we characterize the extremal graphs.

引用次数: 0

A proof of the symmetric theta trees conjecture when q = 0

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.disc.2025.114413

Alessandra Caraceni, Alessandro Iraci

In [7], D'Adderio et al. conjecture a combinatorial formula for the expressions

Ξ e_{α} |_{t = 1}

, known as symmetric Theta trees Conjecture, in terms of tiered trees with an inversion statistic. In [18], Iraci and Romero prove a combinatorial formula for the same symmetric function, in terms of doubly labelled Dyck paths with the area statistic. In this paper, we give an explicit bijection between the subsets of the two families of objects when the relevant statistic is equal to 0, thus proving the symmetric Theta trees conjecture when

q = 0

引用次数: 0

The total vertex irregularity strength for cubic graphs with a perfect matching

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.disc.2025.114402

Aleams Barra , Muhammad Afifurrahman

We compute the exact value of the total vertex irregularity strength of a cubic graph G with a perfect matching. In particular, we confirm that the conjectured value of the total vertex irregularity strength holds. Our method uses a

{1, s}

-edge labeling on the graph G, which can be extended to a vertex irregular labeling on G.

引用次数: 0

Long antipaths and anticycles in oriented graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-28 DOI: 10.1016/j.disc.2025.114412

Bin Chen , Xinmin Hou , Xinyu Zhou

Let

δ^{0} (D)

be the minimum semi-degree of an oriented graph D. Jackson (1981) proved that every oriented graph D with

δ^{0} (D) \geq k

contains a directed path of length 2k when

| V (D) | > 2 k + 2

, and a directed Hamilton cycle when

| V (D) | \leq 2 k + 2

. Stein (2020) further conjectured that every oriented graph D with

δ^{0} (D) > k / 2

contains any orientated path of length k. Recently, Klimošová and Stein (2023) introduced the minimum pseudo-semi-degree

{\tilde{δ}}^{0} (D)

(A slightly weaker variant of the minimum semi-degree condition as

{\tilde{δ}}^{0} (D) \geq δ^{0} (D))

and showed that every oriented graph D with

{\tilde{δ}}^{0} (D) \geq (3 k - 2) / 4

contains each antipath of length k for

k \geq 3

. In this paper, we improve the result of Klimošová and Stein by showing that for all

k \geq 2

, every oriented graph with

{\tilde{δ}}^{0} (D) \geq (2 k + 1) / 3

contains either an antipath of length at least

k + 1

or an anticycle of length at least

k + 1

引用次数: 0

Degree conditions for disjoint path covers in digraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-23 DOI: 10.1016/j.disc.2025.114410

Ansong Ma, Yuefang Sun

In this paper, we study degree conditions for three types of disjoint directed path cover problems: many-to-many k-DDPC, one-to-many k-DDPC and one-to-one k-DDPC, which are intimately connected to other famous topics in graph theory, such as Hamiltonicity and linkage.

We first get two sharp minimum semi-degree sufficient conditions for the unpaired many-to-many k-DDPC problem and a sharp Ore-type degree condition for the paired many-to-many 2-DDPC problem. We then obtain a minimum semi-degree sufficient condition for the one-to-many k-DDPC problem on a digraph with order n, and show that the bound for the minimum semi-degree is sharp when

n + k

is even and is sharp up to an additive constant 1 otherwise. Finally, we give a minimum semi-degree sufficient condition for the one-to-one k-DDPC problem on a digraph with order n, and show that the bound for the minimum semi-degree is sharp when

n + k

is odd and is sharp up to an additive constant 1 otherwise. Furthermore, these results hold even when n is (at least) a linear function of k. In addition, our results improve the existing results by reducing both of the lower bounds of the order and the minimum semi-degree condition of digraphs.

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引用次数: 0

The isomorphism problem of trees from the viewpoint of Terwilliger algebras with respect to an edge

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-23 DOI: 10.1016/j.disc.2025.114407

Shuang-Dong Li , Ying-Ying Tan , Jing Xu , Xiaoye Liang

Let Γ be a finite connected tree, and let

x_{0} x_{0}^{'}

be an edge of Γ. Fix

X_{0} = {x_{0}, x_{0}^{'}}

as the base vertex set, let

Γ^{(X_{0})}

be an edge rooted tree with the edge root

X_{0}

. Let

T = T (X_{0})

be the Terwilliger algebra of

Γ^{(X_{0})}

with respect to

X_{0}

. In this paper, we characterize the structure of the irreducible T-module with the endpoint 0. As a result, it is shown that

T = T (X_{0})

recognizes the isomorphism class of

Γ^{(X_{0})}

引用次数: 0

On the oriented diameter of near planar triangulations

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-23 DOI: 10.1016/j.disc.2025.114406

Yiwei Ge , Xiaonan Liu , Zhiyu Wang

In this paper, we show that the oriented diameter of any n-vertex 2-connected near triangulation is at most

⌈ \frac{n}{2} ⌉

(except for seven small exceptions), and the upper bound is tight. This extends a result of Wang et al. (2021) [29] on the oriented diameter of maximal outerplanar graphs, and improves an upper bound of

n / 2 + O (\sqrt{n})

on the oriented diameter of planar triangulations by Mondal et al. (2024) [24].

引用次数: 0

Three classes of propagation rules for generalized Reed-Solomon codes and their applications to EAQECCs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-22 DOI: 10.1016/j.disc.2025.114405

Ruhao Wan, Shixin Zhu

In this paper, we study the Hermitian hulls of generalized Reed-Solomon (GRS) codes over finite fields. For a given class of GRS codes, by extending the length, increasing the dimension, and extending the length and increasing the dimension at the same time, we obtain three classes of GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore, based on some known

q^{2}

-ary Hermitian self-orthogonal GRS codes with dimension

q - 1

, we obtain several classes of

q^{2}

-ary maximum distance separable (MDS) codes with Hermitian hulls of arbitrary dimensions. It is worth noting that the dimension of these MDS codes can be taken from q to

\frac{n}{2}

, and the parameters of these MDS codes can be more flexible by propagation rules. As an application, we derive three new propagation rules for MDS entanglement-assisted quantum error correction codes (EAQECCs) constructed from GRS codes. Then, from the presently known GRS codes with Hermitian hulls, we can directly obtain many MDS EAQECCs with more flexible parameters. Finally, we present several new classes of (MDS) EAQECCs with flexible parameters, and the distance of these codes can be taken from

q + 1

\frac{n + 2}{2}

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引用次数: 0

Eigenvalues and toughness of regular graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-21 DOI: 10.1016/j.disc.2025.114404

Yuanyuan Chen , Huiqiu Lin , Zhiwen Wang

The toughness of a graph G, denoted by

t (G)

, is defined as

t (G) = \min {\frac{| S |}{c (G - S)} : S \subset V (G) and c (G - S) > 1}

. The bipartite toughness

τ (G)

of a non-complete bipartite graph

G = (X, Y)

is defined as

τ (G) = \min {\frac{| S |}{c (G - S)} : S \subset X or S \subset Y, and c (G - S) > 1}

. Incorporating the toughness and eigenvalues of a graph, we provide two sufficient eigenvalue conditions for a regular graph to be

\frac{1}{b} -

tough for a positive integer b, which extend a significant result by Cioabă and Wong [10]. For a regular bipartite graph, it is proved that

τ (G) \geq 1

. We further show a sufficient eigenvalue condition with the second largest eigenvalue for a regular bipartite graph having bipartite toughness more than 1.

引用次数: 0

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