インサイト - Graph Theory

### Coarse Bottlenecking in Graphs and a Revised Coarse Menger Theorem

This research paper disproves the existing Coarse Menger Conjecture and proposes a revised Coarse Menger-type Theorem based on the concept of coarse bottlenecking in graphs.

### Packing Almost-Spanning Balanced Trees into Complete Bipartite Graphs

For any positive γ, a family of almost-spanning balanced trees (with at most (1-γ)n vertices in each bipartition class) can be packed into a complete bipartite graph Kn,n, provided that the number of trees does not exceed n^(1/2-γ) and n is sufficiently large.

### 반완전 분할 유향 그래프에서 호-분리 내-분지와 외-분지

이 논문에서는 모든 2-호-강 반완전 분할 유향 그래프가 정점 쌍에 관계없이 호-분리 내-분지와 외-분지 쌍을 갖는다는 것을 증명하여 Bang-Jensen과 Wang의 추측을 확인합니다.

### The Existence of Arc-Disjoint In- and Out-Branchings in 2-Arc-Strong Semicomplete Split Digraphs

Every 2-arc-strong semicomplete split digraph contains a good (u, v)-pair (arc-disjoint out-branching and in-branching rooted at u and v respectively) for any choice of vertices u and v, confirming a conjecture by Bang-Jensen and Wang.

### Fractional List Packing for Layered Graphs: Exploring Bounds and Algorithmic Implications

This research paper explores the fractional list packing number (χℓp) and fractional correspondence packing number (χcp) for layered graphs, demonstrating that these parameters can be bounded by graph width measures like treedepth and pathwidth, leading to improved bounds on flexible list coloring.

### Succinct Data Structure for Chordal Graphs with Bounded Vertex Leafage

The authors present a succinct data structure for chordal graphs with bounded vertex leafage that supports efficient adjacency and neighborhood queries.

### Computational Complexity of Recognizing Well-Covered and Generalized Well-Covered Graphs

The core message of this article is to investigate the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs. The authors establish several complexity results, including showing that recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, determining the precise complexity of recognizing 1-extendable (Es) graphs as Θp2-complete, and providing a linear-time algorithm to decide if a chordal graph is 1-extendable.

### Canonical Decompositions of 3-Connected Graphs

Every 3-connected graph can be uniquely decomposed into parts that are either quasi 4-connected, wheels, or thickened K3,m's.

### Tight Lower Bound on the Linear Chromatic Number of Pseudogrids and Improved Relationship Between Centred and Linear Chromatic Numbers

The linear chromatic number of any k × k pseudogrid G is Ω(k), which improves the previously known lower bound and leads to a tighter relationship between the centred chromatic number and the linear chromatic number of graphs.

### Sharp Thresholds for the Existence of Maximum δ-Temporal Cliques in Random Simple Temporal Graphs

The size of a maximum δ-temporal clique in a random simple temporal graph on n vertices is approximately 2 log n / log (1/δ) with high probability.