Journal of Combinatorial Theory Series B最新文献

英文中文

Weak diameter choosability of graphs with an excluded minor

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.005

Joshua Crouch, Chun-Hung Liu

Weak diameter coloring of graphs recently attracted attention, partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph H, H-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring H-minor free graphs with bounded weak diameter. As a corollary, H-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring.

When H is planar, we prove a much stronger result: for every 2-list-assignment L of an H-minor free graph, every precoloring with bounded weak diameter can be extended to an L-coloring with bounded weak diameter. It is a common generalization of earlier results for non-list-coloring with bounded weak diameter and for list-coloring with bounded clustering without allowing precolorings. As a corollary, for any planar graph H and H-minor free graph G, there are exponentially many list-colorings of G with bounded weak diameter (and with bounded clustering if G also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering.

We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree Δ are k-choosable with bounded weak diameter only when

k = Ω (\log Δ / \log \log Δ)

. On the other hand, we show that odd H-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd H-minor free graphs with bounded maximum degree.

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引用次数: 0

Induced C4-free subgraphs with large average degree 具有较大平均度的无 C4 子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.002

Xiying Du , António Girão , Zach Hunter , Rose McCarty , Alex Scott

We prove that there exists a constant C so that, for all

s, k \in N

, if G has average degree at least

k^{C s^{3}}

and does not contain

K_{s, s}

as a subgraph then it contains an induced subgraph which is

C_{4}

-free and has average degree at least k. It was known that some function of s and k suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries.

We show that there exists a constant C so that, for all

s, k \in N

, if G has average degree at least

k^{C s^{3}}

and does not contain

K_{s, s}

as a subgraph then it contains an induced subdivision of

K_{k}

. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both k and s.

We also show that for any hereditary degree-bounded class

F

, there exists a constant

C = C_{F}

so that

C^{s^{3}}

is a degree-bounding function for

F

. This is the first bound of any type on the rate of growth of such functions.

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引用次数: 0

A matrix realization of spectral bounds

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.006

Yen-Jen Cheng , Chih-wen Weng

We give a unified and systematic way to find bounds for the largest real eigenvalue of a nonnegative matrix by considering its modified quotient matrix. We leverage this insight to identify the unique matrix whose largest real eigenvalue is maximum among all

(0, 1)

-matrices with a specified number of ones. This result resolves a problem that was posed independently by R. Brualdi and A. Hoffman, as well as F. Friedland, back in 1985.

引用次数: 0

Pivot-minors and the Erdős-Hajnal conjecture

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-16 DOI: 10.1016/j.jctb.2025.04.004

James Davies

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph H, there exists

ϵ > 0

such that every n-vertex graph with no pivot-minor isomorphic to H contains two sets

A, B

of vertices such that

| A |, | B | ⩾ ϵ n

and A is complete or anticomplete to B.

引用次数: 0

Haar graphical representations of finite groups and an application to poset representations

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-16 DOI: 10.1016/j.jctb.2025.04.001

Joy Morris , Pablo Spiga

Let R be a group and let S be a subset of R. The Haar graph

Haar (R, S)

of R with connection set S is the graph having vertex set

R \times {- 1, 1}

, where two distinct vertices

(x, - 1)

and

(y, 1)

are declared to be adjacent if and only if

y x^{- 1} \in S

. The name Haar graph was coined by Tomaž Pisanski in one of the first investigations on this class of graphs.

For every

g \in R

, the mapping

ρ_{g} : (x, ε) \mapsto (x g, ε)

\forall (x, ε) \in R \times {- 1, 1}

, is an automorphism of

Haar (R, S)

. In particular, the set

\hat{R} : = {ρ_{g} | g \in R}

is a subgroup of the automorphism group of

Haar (R, S)

isomorphic to R. In the case that the automorphism group of

Haar (R, S)

equals

\hat{R}

, the Haar graph

Haar (R, S)

is said to be a Haar graphical representation of the group R.

Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions.

Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.

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引用次数: 0

Optimal bounds for zero-sum cycles. I. Odd order 零和循环的最优边界。I. 奇数阶

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-16 DOI: 10.1016/j.jctb.2025.04.003

Rutger Campbell , J. Pascal Gollin , Kevin Hendrey , Raphael Steiner

For a finite (not necessarily abelian) group

(Γ, \cdot)

, let

n (Γ)

denote the smallest positive integer n such that for each labelling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich [2] initiated the study of the parameter

n (\cdot)

on cyclic groups and proved

n (Z_{q}) = O (q \log q)

. This was later improved to a linear bound of

n (Γ) \leq 8 | Γ |

for every finite abelian group by Mészáros and the last author [8], and then further to

n (Γ) \leq 2 | Γ | - 1

for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma [3] as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta [1].

In this series of two papers we conclude this line of research by proving that

n (Γ) \leq | Γ | + 1

for every finite group

(Γ, \cdot)

, which is the best possible such bound in terms of the group order and precisely determines the value of

n (Γ)

for all cyclic groups as

n (Z_{q}) = q + 1

In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.

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引用次数: 0

A splitter theorem on 3-connected binary matroids and inner fans

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-01 DOI: 10.1016/j.jctb.2025.03.004

João Paulo Costalonga

We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids

N < M

, there is a disjoint family

{X_{1}, \dots, X_{n}} \subseteq 2^{E (M)}

such that

r (X_{1}) + \dots + r (X_{n}) = r (X_{1} \cup \dots \cup X_{n}) \geq r (M) - r (N)

, each

si (M / X_{i})

is 3-connected with an N-minor, and either

| X_{i} | = 1

or X is a special type of fan. We also establish a stronger version of this result under specific hypotheses. These results have several consequences, including the generalizations for binary matroids of some results about contractible edges in 3-connected graphs and some other structural results for graphs and binary matroids.

引用次数: 0

Sparse induced subgraphs of large treewidth

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-03-20 DOI: 10.1016/j.jctb.2025.03.002

Édouard Bonnet

Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour (1986) [22] or by a classic result of Chekuri and Chuzhoy (2015) [5], we show that for any natural numbers t and w, and real

ε > 0

, there is an integer

W : = W (t, w, ε)

such that every graph with treewidth at least W and no

K_{t, t}

subgraph admits a 2-connected n-vertex induced subgraph with treewidth at least w and at most

(1 + ε) n

edges. The induced subgraph is either a subdivided wall, or its line graph, or a spanning supergraph of a subdivided biclique. This in particular extends a result of Weißauer (2019) [25] that graphs of large treewidth have a large biclique subgraph or a long induced cycle.

{"title":"Sparse induced subgraphs of large treewidth","authors":"Édouard Bonnet","doi":"10.1016/j.jctb.2025.03.002","DOIUrl":"10.1016/j.jctb.2025.03.002","url":null,"abstract":"<div><div>Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour (1986) <span><span>[22]</span></span> or by a classic result of Chekuri and Chuzhoy (2015) <span><span>[5]</span></span>, we show that for any natural numbers <em>t</em> and <em>w</em>, and real <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, there is an integer <span><math><mi>W</mi><mo>:</mo><mo>=</mo><mi>W</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>ε</mi><mo>)</mo></math></span> such that every graph with treewidth at least <em>W</em> and no <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> subgraph admits a 2-connected <em>n</em>-vertex induced subgraph with treewidth at least <em>w</em> and at most <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo><mi>n</mi></math></span> edges. The induced subgraph is either a subdivided wall, or its line graph, or a spanning supergraph of a subdivided biclique. This in particular extends a result of Weißauer (2019) <span><span>[25]</span></span> that graphs of large treewidth have a large biclique subgraph or a long induced cycle.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 184-203"},"PeriodicalIF":1.2,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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

Some results and problems on tournament structure

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-28 DOI: 10.1016/j.jctb.2025.02.002

Tung Nguyen , Alex Scott , Paul Seymour

This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments

A, B

, both with large chromatic number, such that all edges between them are directed from A to B? We describe what we know about this question, and report some progress on several other related questions, on tournament colouring and domination.

引用次数: 0

Ramsey numbers of bounded degree trees versus general graphs

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-21 DOI: 10.1016/j.jctb.2025.02.004

Richard Montgomery , Matías Pavez-Signé , Jun Yan

For every

k \geq 2

and Δ, we prove that there exists a constant

C_{Δ, k}

such that the following holds. For every graph H with

χ (H) = k

and every tree T with at least

C_{Δ, k} | H |

vertices and maximum degree at most Δ, the Ramsey number

R (T, H)

(k - 1) (| T | - 1) + σ (H)

, where

σ (H)

is the size of a smallest colour class across all proper k-colourings of H. This is tight up to the value of

C_{Δ, k}

, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.

引用次数: 0

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Journal of Combinatorial Theory Series B

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