Core Concepts

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.

Abstract

The paper studies the relationship between the centred chromatic number (χcen(G)) and the linear chromatic number (χlin(G)) of graphs. The centred chromatic number is equivalent to the treedepth of a graph and has been extensively studied, while the linear chromatic number is less well understood.
The main contributions are:
Establishing a tight lower bound on the linear chromatic number of pseudogrids: For any k × k pseudogrid G, χlin(G) ∈Ω(k). This improves the previously known lower bound of Ω(√k).
Using this tight lower bound, the authors improve the general upper bound relating the centred and linear chromatic numbers from χcen(G) ≤(χlin(G))19 · (log(χlin(G)))O(1) to χcen(G) ≤(χlin(G))10 · (log(χlin(G)))O(1).
The tight lower bound on the linear chromatic number of pseudogrids provides further evidence supporting the conjecture that the centred chromatic number (i.e., the treedepth) of any graph is upper bounded by a linear function of its linear chromatic number.
The proof of the tight lower bound on the linear chromatic number of pseudogrids is technical and involves several steps, including:
Removing rows and columns from the pseudogrid to ensure each colour appears frequently.
Constructing a well-separated set of vertices that contains two vertices of each colour.
Assembling a path that contains all vertices in the well-separated set.

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Deeper Inquiries

The techniques used in this paper to establish tight lower bounds on the linear chromatic number of pseudogrids can be extended to other graph classes by adapting the concept of well-separated sets and the packing lemma. For different graph classes, one would need to define appropriate criteria for well-separated pairs or sets of vertices and modify the packing lemma accordingly. Additionally, the application of the Local Lemma and probabilistic methods to ensure the existence of certain structures in the graph could be generalized to other graph classes with similar properties. By carefully analyzing the specific characteristics of the graph class in question, one can tailor the approach to establish tight lower bounds on the linear chromatic number.

The improved relationship between centred and linear chromatic numbers, as demonstrated in this paper, has significant implications for practical applications that rely on efficiently computable approximations of treedepth. Treedepth is a fundamental parameter in graph theory that has applications in various fields such as algorithm design, network analysis, and computational biology. By establishing a tighter bound on the linear chromatic number in relation to centred chromatic number, it provides a more accurate approximation of treedepth for a given graph. This can lead to more efficient algorithms, better network modeling, and improved understanding of complex biological systems represented as graphs.

The linear chromatic number of a graph is closely related to other graph parameters, offering potential insights into the structure of graphs. One interesting connection is with the tree-depth of a graph, as highlighted in the paper. The linear chromatic number is equivalent to treedepth, which measures how "tree-like" a graph is. Exploring the relationship between linear chromatic number and other parameters such as chromatic number, clique number, or girth could reveal further structural properties of graphs. Additionally, investigating the computational complexity of determining the linear chromatic number and its implications on graph coloring algorithms could lead to advancements in graph theory and algorithm design.

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