Let K = K(a, p;λ1, λ2) be the multigraph with: the number of vertices in each part equal to a; the number of parts equal to p; the number of edges joining any two vertices of the same part equal to λ1; and the number of edges joining any two vertices of different parts equal to λ2. This graph was of interest to Bose and Shimamoto in their study of group divisible designs with two associate clas...

A graph is 1-planar if it can be embedded in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree 5 and girth 4 contains (1) a 5-vertex adjacent to an ≤ 6-vertex, (2) a 4-cycle whose every vertex has degree at most 9, (3) a K1,4 with all vertices having degree at most 11.

BACKGROUND The renewal of removable dentures is often suggested to denture wearers subject to discomfort. However, the impact of this rehabilitation on patients' oral health related quality of life and their removable dentures related satisfaction is still unknown. This study was aimed at assessing these patient-centered outcomes and the potential impact of different factors. METHODS A cohort...

The development of catalytic tools for the late-stage modification amino acids within a peptide framework is challenging task capital importance. Herein, we report Ru-catalyzed C(sp2)?H hydroxylation collection Tyr-containing di- and tripeptides featuring use carbamate as removable directing group PhI(OCOCF3)2 (PIFA) oxidant. This air-compatible tagging technique reliable, scalable provides acc...

let g be a graph with p vertices and q edges and a = {0, 1, 2, . . . , [q/2]}. a vertex labeling f : v (g) → a induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. for a ∈ a, let vf (a) be the number of vertices v with f(v) = a. a graph g is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the in...

The classic K-Cycle problem asks if a graph G, with vertex set V (G), has a simple cycle containing all vertices of a given set K TM V (G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K TM V (G) and an injective coloring c : K æ {1, 2, . . . , |K|}, decide if G has a simple cycle containing each color in {1, 2, . . . , |K|} (once). Another problem widely k...

The notion of removable classes arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. The last (single or double) ear of an ear decomposition is defined as a removable class. Every matching covered graph not induced by a circuit has at least three removable classes. In this paper, we characterize matching covered graphs with precisely three re...

A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. We show that a feedback vertex set approximating a minimum one within a constant factor can be e ciently found in undirected graphs. In fact the derived approximation ratio matches the best constant ratio known today for the vertex cover problem, improving the previous best ...

A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. We show that a feedback vertex set approximating a minimum one within a constant factor can be e ciently found in undirected graphs. In fact the derived approximation ratio matches the best constant ratio known today for the vertex cover problem, improving the previous best ...