Journal of Combinatorial Theory Series B最新文献

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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

Tree amalgamations and quasi-isometries

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-14 DOI: 10.1016/j.jctb.2025.02.003

Matthias Hamann

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papasoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.

引用次数: 0

A counterexample to the coarse Menger conjecture

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-13 DOI: 10.1016/j.jctb.2025.01.004

Tung Nguyen , Alex Scott , Paul Seymour

Menger's well-known theorem from 1927 characterizes when it is possible to find k vertex-disjoint paths between two sets of vertices in a graph G. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the k paths are required to be pairwise at some distance at least d. The result is known for

k \leq 2

, but we will show that it is false for all

k \geq 3

, even if G is constrained to have maximum degree at most three. We also give a simpler proof of the result when

k = 2

引用次数: 0

Clustered coloring of (path + 2K1)-free graphs on surfaces

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-13 DOI: 10.1016/j.jctb.2025.02.001

Zdeněk Dvořák

Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let

P_{t}^{″}

be the complete join of two isolated vertices with a path on t vertices. For any surface Σ, a subgraph-closed class of graphs drawn on Σ is 3-choosable with bounded clustering if and only if there exists t such that

P_{t}^{″}

does not belong to the class.

引用次数: 0

Ascending subgraph decomposition

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-12 DOI: 10.1016/j.jctb.2025.01.003

Kyriakos Katsamaktsis , Shoham Letzter , Alexey Pokrovskiy , Benny Sudakov

A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph G into copies of

H_{1}, \dots, H_{m}

are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with

(\begin{matrix} m + 1 \\ 2 \end{matrix})

edges can be decomposed into subgraphs

H_{1}, \dots, H_{m}

such that each

H_{i}

has i edges and is isomorphic to a subgraph of

H_{i + 1}

. In this paper we prove this conjecture for sufficiently large m.

引用次数: 0

Every d(d + 1)-connected graph is globally rigid in Rd

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-12 DOI: 10.1016/j.jctb.2025.01.005

Soma Villányi

Using a probabilistic method, we prove that

d (d + 1)

-connected graphs are rigid in

R^{d}

, a conjecture of Lovász and Yemini. Then, using recent results on weakly globally linked pairs, we modify our argument to prove that

d (d + 1)

-connected graphs are globally rigid, too, a conjecture of Connelly, Jordán and Whiteley. The constant

d (d + 1)

is best possible.

引用次数: 0

Cumulant expansion for counting Eulerian orientations

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-24 DOI: 10.1016/j.jctb.2025.01.002

Mikhail Isaev , Brendan D. McKay , Rui-Ray Zhang

An Eulerian orientation is an orientation of the edges of a graph such that every vertex is balanced: its in-degree equals its out-degree. Counting Eulerian orientations corresponds to the crucial partition function in so-called “ice-type models” in statistical physics and is known to be hard for general graphs. For all graphs with good expansion properties and degrees larger than

\log^{8} n

, we derive an asymptotic expansion for this count that approximates it to precision

O (n^{- c})

for arbitrarily large c, where n is the number of vertices. The proof relies on a new tail bound for the cumulant expansion of the Laplace transform, which is of independent interest.

引用次数: 0

Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-23 DOI: 10.1016/j.jctb.2025.01.001

Jianfeng Hou , Caiyun Hu , Heng Li , Xizhi Liu , Caihong Yang , Yixiao Zhang

Given an r-graph F with

r \geq 2

, let

ex (n, (t + 1) F)

denote the maximum number of edges in an n-vertex r-graph with at most t pairwise vertex-disjoint copies of F. Extending several old results and complementing prior work [34] on nondegenerate hypergraphs, we initiate a systematic study on

ex (n, (t + 1) F)

for degenerate hypergraphs F.

For a broad class of degenerate hypergraphs F, we present near-optimal upper bounds for

ex (n, (t + 1) F)

when n is sufficiently large and t lies in intervals

[0, \frac{ε \cdot ex (n, F)}{n^{r - 1}}]

[\frac{ex (n, F)}{ε n^{r - 1}}, ε n]

, and

[(1 - ε) \frac{n}{v (F)}, \frac{n}{v (F)}]

, where

ε > 0

is a constant depending only on F. Our results reveal very different structures for extremal constructions across the three intervals, and we provide characterizations of extremal constructions within the first interval. Additionally, we characterize extremal constructions within the second interval for graphs. Our proof for the first interval also applies to a special class of nondegenerate hypergraphs, including those with undetermined Turán densities, partially improving a result in [34].

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引用次数: 0

The next case of Andrásfai's conjecture

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-14 DOI: 10.1016/j.jctb.2024.12.010

Tomasz Łuczak , Joanna Polcyn , Christian Reiher

Let

ex (n, s)

denote the maximum number of edges in a triangle-free graph on n vertices which contains no independent sets larger than s. The behaviour of

ex (n, s)

was first studied by Andrásfai, who conjectured that for

s > n / 3

this function is determined by appropriately chosen blow-ups of so called Andrásfai graphs. Moreover, he proved

ex (n, s) = n^{2} - 4 n s + 5 s^{2}

for

s / n \in [2 / 5, 1 / 2]

and in earlier work we obtained

ex (n, s) = 3 n^{2} - 15 n s + 20 s^{2}

for

s / n \in [3 / 8, 2 / 5]

. Here we make the next step in the quest to settle Andrásfai's conjecture by proving

ex (n, s) = 6 n^{2} - 32 n s + 44 s^{2}

for

s / n \in [4 / 11, 3 / 8]

引用次数: 0

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