A (vertex) ℓ-ranking is a labelling ϕ : V (G) → N of the vertices of a graph G with integer colours so that for any path u 0 , . . . , u p of length at most ℓ, ϕ(u 0 ) ϕ(u p ) or ϕ(u 0 ) < max{ϕ(u 0 ), . . . , ϕ(u p )}. We show that, for any fixed integer ℓ ≥ 2, every n-vertex planar graph has an ℓ-ranking using O(log n/ log log log n) colours and this is tight even when ℓ = 2; for infinitely many values of n, there are n-vertex planar graphs, for which any 2-ranking requires Ω(log n/ log log log n) colours. This result also extends to bounded genus graphs.In developing this proof we obtain optimal bounds on the number of colours needed for ℓ-ranking graphs of treewidth t and graphs of simple treewidth t. These upper bounds are constructive and give O(n log n)-time algorithms. Additional results that come from our techniques include new sublogarithmic upper bounds on the number of colours needed for ℓ-rankings of apex minor-free graphs and k-planar graphs.

Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer $k$ we describe a graph with layered treewidth 1 and row treewidth $k$. We also prove an analogous result for layered pathwidth and row pathwidth.

The undirected power graph of a finite group G, P (G), is a graph with the group elements of G as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let A be an adjacency matrix of P (G). An eigenvalue λ of A is a main eigenvalue if the eigenspace ε(λ) has an eigenvector X such that X t j ̸ = 0, where j is the all-one vector.In this paper we want to focus on the power graph of the finite cyclic group Zn and find a condition on n where P (Zn) has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of P (Zn) where n has a unique prime decomposition n = p r p2. We also formulate a conjecture on the number of the main eigenvalues of P (Zn) for an arbitrary positive integer n.

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We prove that the linear chromatic number of any k × k pseudogrid is Ω(k). By an argument of Kun et al (Algorithmica, 2021), this result gives a tighter upper bound on the treedepth of a graph as a function of its linear chromatic number and gives further evidence in support of their conjecture that the treedepth of any graph is upper bounded by a linear function of its linear chromatic number.