منابع مشابه
Diameter bounds for planar graphs
The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(|V | − 1) − |E|)/3 and 4|V |2/3|E| on the diameter (for connected planar graphs), w...
متن کاملImproving diameter bounds for distance-regular graphs
In this paper we study the sequence (ci)0≤i≤d for a distance-regular graph. In particular we show that if d ≥ 2j and cj = c > 1 then c2j > c holds. Using this we give improvements on diameter bounds by Hiraki and Koolen [5], and Pyber [8], respectively, by applying this inequality.
متن کاملClass 1 bounds for planar graphs
There are planar class 2 graphs with maximum vertex-degree k, for each k ∈ {2, 3, 4, 5}. In 1965, Vizing proved that every planar graph with ∆ ≥ 8 is class 1. He conjectured that every planar graph with ∆ ≥ 6 is a class 1 graph. This conjecture is proved for ∆ = 7, and still open for ∆ = 6. Let k ≥ 2 and G be a k-critical planar graph. The average face-degree F (G) of G is 2 |F (G)| |E(G)|. Let...
متن کاملFace-Degree Bounds for Planar Critical Graphs
The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs, based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters...
متن کاملTight Bounds for Linkages in Planar Graphs
The Disjoint-Paths Problem asks, given a graph G and a set of pairs of terminals (s1, t1), . . . , (sk, tk), whether there is a collection of k pairwise vertex-disjoint paths linking si and ti, for i = 1, . . . , k. In their f(k) · n algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2011
ISSN: 0012-365X
DOI: 10.1016/j.disc.2010.11.006