Discrete Mathematics最新文献

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On the nontrivial extremal eigenvalues of graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-11 DOI: 10.1016/j.disc.2025.114423

Wenbo Li, Shiping Liu

We present improved bounds on a quantitative version of an observation originally due to Breuillard, Green, Guralnick and Tao which says that for finite non-bipartite Cayley graphs, once the nontrivial eigenvalues of their normalized adjacency matrices are uniformly bounded away from 1, then they are also uniformly bounded away from −1. Unlike previous works which depend heavily on combinatorial arguments, we rely more on analysis of eigenfunctions. We establish a new explicit lower bound for the gap between −1 and the smallest normalized adjacency eigenvalue, which improves previous lower bounds in terms of edge-expansion, and is comparable to the best known lower bound in terms of vertex-expansion.

引用次数: 0

Multigraph edge-coloring with local list sizes

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-07 DOI: 10.1016/j.disc.2025.114420

Abhishek Dhawan

Let G be a multigraph and

L : E (G) \to 2^{N}

be a list assignment for the edges of G. Suppose additionally, for every vertex x, the edges incident to x have at least

f (x)

colors in common. We consider a variant of local edge-colorings wherein the color received by an edge e must be contained in

L (e)

. The locality appears in the function f, i.e.,

f (x)

is some function of the local structure of x in G. Such a notion is a natural generalization of traditional local edge-coloring. Our main results include sufficient conditions on the function f to construct such colorings. As corollaries, we obtain local analogs of Vizing and Shannon's theorems, recovering a recent result of Conley, Grebík, and Pikhurko.

引用次数: 0

High-order spectral characterizations of graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-07 DOI: 10.1016/j.disc.2025.114421

Lixiang Chen , Lizhu Sun , Changjiang Bu

The tensor spectra of the power hypergraph of a graph G is called the high-order spectra of G. In this paper, we show that all Smith graphs are determined by their high-order spectra. We give some high-order cospectral invariants of trees and use them to show that some cospectral trees constructed by the classical Schwenk method can be distinguished by their high-order spectra.

引用次数: 0

On the saturation spectrum of the unions of disjoint cycles

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-07 DOI: 10.1016/j.disc.2025.114418

Yue Ma

Let G be a graph and

H

be a family of graphs. We say G is

H

-saturated if G does not contain a copy of H with

H \in H

, but the addition of any edge

e \notin E (G)

creates at least one copy of some

H \in H

within

G + e

. The saturation number of

H

is the minimum size of an

H

-saturated graph on n vertices, and the saturation spectrum of

H

is the set of all possible sizes of an

H

-saturated graph on n vertices. Let

k C_{\geq 3}

be the family of the unions of k vertex-disjoint cycles. In this note, we completely determine the saturation number and the saturation spectrum of

k C_{\geq 3}

for

k = 2

and give some results for

k \geq 3

{"title":"On the saturation spectrum of the unions of disjoint cycles","authors":"Yue Ma","doi":"10.1016/j.disc.2025.114418","DOIUrl":"10.1016/j.disc.2025.114418","url":null,"abstract":"<div><div>Let G be a graph and <math><mi>H</mi></math> be a family of graphs. We say G is <math><mi>H</mi></math>-saturated if G does not contain a copy of H with <math><mi>H</mi><mo>∈</mo><mi>H</mi></math>, but the addition of any edge <math><mi>e</mi><mo>∉</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> creates at least one copy of some <math><mi>H</mi><mo>∈</mo><mi>H</mi></math> within <math><mi>G</mi><mo>+</mo><mi>e</mi></math>. The saturation number of <math><mi>H</mi></math> is the minimum size of an <math><mi>H</mi></math>-saturated graph on n vertices, and the saturation spectrum of <math><mi>H</mi></math> is the set of all possible sizes of an <math><mi>H</mi></math>-saturated graph on n vertices. Let <math><mi>k</mi><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mn>3</mn></mrow></msub></math> be the family of the unions of k vertex-disjoint cycles. In this note, we completely determine the saturation number and the saturation spectrum of <math><mi>k</mi><msub><mrow><mi>C</mi></mrow><mrow><mo>≥</mo><mn>3</mn></mrow></msub></math> for <math><mi>k</mi><mo>=</mo><mn>2</mn></math> and give some results for <math><mi>k</mi><mo>≥</mo><mn>3</mn></math>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 6","pages":"Article 114418"},"PeriodicalIF":0.7,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A spanning tree with at most k leaves in a K1,5-free graph

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-07 DOI: 10.1016/j.disc.2025.114411

Pei Sun , Yuan Chen , Masahiro Kimura , Kenta Ozeki , Masao Tsugaki

A tree is called a k-ended tree if it has at most k leaves. Let

k \geq 2

and

p \geq 3

be integers, let G be a connected

K_{1, p}

-free graph, and let

σ_{k + 1} (G)

be the minimum degree sum of pair-wisely non-adjacent

k + 1

vertices of G. For

p = 3, 4

or for

p = 5

and

k = 4, 5, 6

, the lower bounds of

σ_{k + 1} (G)

which assure the existence of spanning k-ended trees are known. In this paper, we extend these results to the case

p = 5

and any

k \geq 2

, which states that for a connected

K_{1, p}

-free graph, if

k \geq 4

and

σ_{k + 1} (G) \geq | G | - k / 3

, or if

k = 3

and

σ_{k + 1} (G) \geq | G |

, or if

k = 2

| G | \geq 7

and

σ_{k + 1} (G) \geq | G |

, then G has a spanning k-ended tree. These lower bounds of the assumptions are best possible.

引用次数: 0

Transformations of 2-port networks and tiling by rectangles

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-02-05 DOI: 10.1016/j.disc.2025.114419

Svetlana Shirokovskikh

This paper presents a novel investigation into the properties of 2-port networks, introducing the concepts of voltage drop and Π-equivalence. The primary contribution is the demonstration that any planar network is Π-equivalent to a network with a maximum of five edges. This finding has significant implications for tiling problems, specifically in relation to octagons shaped like the letter Π. We establish that if such an octagon can be tiled by squares, it can also be tiled by no more than five rectangles with rational aspect ratios. The theorem by Y. C. de Verdière, I. Gitler, and D. Vertigan from 1996 proves this only for 6 rectangles. In our approach, we use Π-equivalent transformations to simplify the network's structure. A novel transformation, which we have named Box-H, plays a crucial role in this process. By applying these transformations, we are able to significantly reduce the complexity of 2-port networks.

引用次数: 0

Corrigendum to “Combinatorial Nullstellensatz and Turán numbers of complete r-partite r-uniform hypergraphs” [Discrete Math. 347 (2024) 114037]

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-31 DOI: 10.1016/j.disc.2025.114417

Alexey Gordeev

We correct the statement of Lemma 2.1 in [2] and give a new proof of Corollary 2.2 which previously depended on the incorrect version of Lemma 2.1.

引用次数: 0

Shift graphs, chromatic number and acyclic one-path orientations

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-31 DOI: 10.1016/j.disc.2025.114414

Arpan Sadhukhan

Shift graphs, which were introduced by Erdős and Hajnal [9], [11], have been used to answer various questions in structural graph theory. In this paper, we prove two new results using shift graphs and their induced subgraphs.

Recently Girão et al. [13], showed that for every graph F with at least one edge, there is a constant $c_{F}$ such that there are graphs of arbitrarily large chromatic number and the same clique number as F, in which every F-free induced subgraph has chromatic number at most $c_{F}$ . We significantly improve the value of the constant $c_{F}$ for the special case where F is the complete bipartite graph $K_{a, b}$ . We show that any $K_{a, b}$ -free induced subgraph of the triangle-free shift graph $G_{n, 2}$ has chromatic number bounded by $O (\log (a + b))$ .
An undirected simple graph G is said to have the AOP Property if it can be acyclically oriented such that there is at most one directed path between any two vertices. We prove that the shift graph $G_{n, 2}$ does not have the AOP property for any $n ⩾ 9$ . Despite this, we construct induced subgraphs of shift graph $G_{n, 2}$ with an arbitrarily high chromatic number and odd-girth that have the AOP property.

Furthermore, we construct graphs with arbitrarily high odd-girth that do not have the AOP Property and also prove the existence of graphs with girth equal to 5 that do not have the AOP property.

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引用次数: 0

On the maximum second eigenvalue of outerplanar graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-31 DOI: 10.1016/j.disc.2025.114416

George Brooks , Maggie Gu , Jack Hyatt , William Linz , Linyuan Lu

For a fixed positive integer k and a graph G, let

λ_{k} (G)

denote the k-th largest eigenvalue of the adjacency matrix of G. In 2017, Tait and Tobin [24] proved that the maximum

λ_{1} (G)

among all outerplanar graphs on n vertices is achieved by the fan graph

K_{1} \lor P_{n - 1}

. In this paper, we consider a similar problem of determining the maximum

λ_{2}

among all connected outerplanar graphs on n vertices. For n even and sufficiently large, we prove that the maximum

λ_{2}

is uniquely achieved by the graph

(K_{1} \lor P_{n / 2 - 1}) - (K_{1} \lor P_{n / 2 - 1})

, which is obtained by connecting two disjoint copies of

(K_{1} \lor P_{n / 2 - 1})

through a new edge joining their smallest degree vertices. When n is odd and sufficiently large, the extremal graphs are not unique. The extremal graphs are those graphs G that contain a cut vertex u such that

G ∖ {u}

is isomorphic to

2 (K_{1} \lor P_{n / 2 - 1})

. We also determine the maximum

λ_{2}

among all 2-connected outerplanar graphs and asymptotically determine the maximum of

λ_{k} (G)

among all connected outerplanar graphs for any fixed k.

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引用次数: 0

The number of cliques in hypergraphs with forbidden subgraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-01-29 DOI: 10.1016/j.disc.2025.114415

Ayush Basu , Vojtěch Rödl , Yi Zhao

We study the maximum number of r-vertex cliques in

(r - 1)

-uniform hypergraphs not containing complete r-partite hypergraphs

K_{r}^{(r - 1)} (a_{1}, \dots, a_{r})

. By using the hypergraph removal lemma, we show that this maximum is

o (n^{r - 1 / (a_{1} \dots a_{r - 1})})

. This immediately implies the corresponding results of Mubayi and Mukherjee and of Balogh, Jiang, and Luo for graphs. We also provide a lower bound by using hypergraph Turán numbers.

引用次数: 0

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