Abstract Let X be an n-element point set in the k-dimensional unit cube $$[0,1]^k$$ [ 0 , 1 ] k where $$k \ge 2$$ k ≥ 2 . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour) $$x_1, x_2, \ldots , x_n$$ x 1 , x 2 , … , x n through the n points, such that $$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ∑ i = 1 n | x i - x i + 1 | k 1 / k ≤ c k , where $$|x-y|$$ | x - y | is the Euclidean distance between x and y, and $$c_k$$ c k is an absolute constant that depends only on k, where $$x_{n+1} \equiv x_1$$ x n + 1 ≡ x 1 . From the other direction, for every $$k \ge 2$$ k ≥ 2 and $$n \ge 2$$ n ≥ 2 , there exist n points in $$[0,1]^k$$ [ 0 , 1 ] k , such that their shortest tour satisfies $$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ∑ i = 1 n | x i - x i + 1 | k 1 / k = 2 1 / k · k . For the plane, the best constant is

Abstract. Denote by [Formula: see text] the smallest eigenvalue of the signless Laplacian matrix of an [Formula: see text]-vertex graph [Formula: see text]. Brandt conjectured in 1997 that for regular triangle-free graphs [Formula: see text]. We prove a stronger result: If [Formula: see text] is a triangle-free graph, then [Formula: see text]. Brandt’s conjecture is a subproblem of two famous conjectures of Erdős: (1) Sparse-half-conjecture: Every [Formula: see text]-vertex triangle-free graph has a subset of vertices of size [Formula: see text] spanning at most [Formula: see text] edges. (2) Every [Formula: see text]-vertex triangle-free graph can be made bipartite by removing at most [Formula: see text] edges. In our proof we use linear algebraic methods to upper bound [Formula: see text] by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.

A graph where each vertex v has a list L(v) of available colors is L-colorable if there is a proper coloring such that the color of v is in L(v) for each v. A graph is k-choosable if every assignment L of at least k colors to each vertex guarantees an L-coloring. Given a list assignment L, an L-request for a vertex v is a color c∈L(v). In this paper, we look at a variant of the widely studied class of precoloring extension problems from Dvořák, Norin, and Postle (J. Graph Theory, 2019), wherein one must satisfy “enough”, as opposed to all, of the requested set of precolors. A graph G is ɛ-flexible for list size k if for any k-list assignment L, and any set S of L-requests, there is an L-coloring of G satisfying ɛ-fraction of the requests in S. It is conjectured that planar graphs are ɛ-flexible for list size 5, yet it is proved only for list size 6 and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masařík and show that planar graphs without K4− are ɛ-flexible for list size 5. We also prove that planar graphs without 4-cycles and 3-cycle distance at least 2 are ɛ-flexible for list size 4. Finally, we introduce a new (slightly weaker) form of ɛ-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are ɛ-flexible for list size 5.