We show that there exists an adjacency labelling scheme for planar graphs where each vertex of n -vertex graph G is assigned a (1 + o(1)) log 2 -bit label and the labels two vertices u v are sufficient to determine if uv edge . This optimal up lower order term first such asymptotically result. An alternative, but equivalent, interpretation this result that, every positive integer , U with 1+o(1...

A k-tree is a tree with maximum degree at most k, and a k-walk is a closed walk with each vertex repeated at most k times. A k-walk can be obtained from a k-tree by visiting each edge of the k-tree twice. Jackson and Wormald conjectured in 1990 that for k ≥ 2, every 1 k−1 -tough connected graph contains a spanning k-walk. This conjecture is open even for planar graphs. We confirm this conjectur...

The decays KL → γγ and KL → l+l− γ are studied at the leading order p6 in Chiral Perturbation Theory. One-loop contributions stemming from the odd intrinsic parity | ∆S |= 1 effective Lagrangian of order p4 are included and shown to be of possible relevance. They affect the decay KL → γγ adding to the usual pole terms a piece free of counterterm uncertainties. In the case of the KL → l+l− γ dec...

The domination number of a graph G = (V,E) is the minimum size of a dominating set U ⊆ V , which satisfies that every vertex in V \ U is adjacent to at least one vertex in U . The notion of a problem kernel refers to a polynomial time algorithm that achieves some provable reduction of the input size. Given a graph G whose domination number is k, the objective is to design a polynomial time algo...

An infinite binary sequence X is Kolmogorov-Loveland (or KL) random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X . A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence. One of the major open ...

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed g and k, we show that every n-vertex graph that can be embedded on a surface of genus g with at most k cro...

Abstract We determine the maximum number of edges in a $$K_4$$ K 4 -minor-free n -vertex graph girth g , when $$g=5$$ g = 5 or is even. argue that there are many different extremal graphs if even ...

In this paper we introduce a natural generalization of the well-known problems Cluster Editing and Bicluster Editing, whose parameterized versions have been intensively investigated in the recent literature. The generalized problem, called Mixed Cluster Editing or M-Cluster Editing, is formulated as follows. Let M be a family of graphs. Given a graph G and a nonnegative integer k, transform G, ...

Let H be a xed graph. We show that any H-minor free graph of high enough girth has a circular-chromatic number arbitrarily close to two. Equivalently, such graphs have homomorphisms into a large odd circuit. In particular, graphs of high girth and of bounded genus or bounded tree width are \nearly bipartite" in this sense. For example, any planar graph of girth at least 16 admits a homomorphism...

BACKGROUND Recent clinical studies have suggested that serum surfactant protein (SP) A, SP-D and Krebs von den Lungen-6 (KL-6) are potential biomarkers for interstitial lung diseases. Serum levels of SP-A and SP-D have also been found to be elevated in chronic obstructive pulmonary disease (COPD), but their significance has not been evaluated or compared in induced sputum samples obtained direc...