146,308 research outputs found
Forced color classes, intersection graphs and the strong perfect graph conjecture
AbstractIn 1996, A. Sebő[11] raised the following two conjectures concerned with the famous Strong Perfect Graph Conjecture: (1) Suppose that a minimally imperfect graph G has a vertex p incident to 2ω(G)−2 determined edges and that its complement Ḡ has a vertex q incident to 2α(G)−2 determined edges. (An edge of G is called determined if an ω-clique of G contains both of its endpoints.) Then G is an odd hole or an odd antihole. (2) Let v0 be a vertex of a partitionable graph G. And suppose A,B to be ω-cliques of G so that v0∈A∩B. If every ω-clique K containing the vertex v0 is contained in A∪B, then G is an odd hole or an odd antihole. In this paper, we will prove (1) for a minimally imperfect graph G such that (p,q) is a determined edge of either G or Ḡ, and prove (2) for a minimally imperfect graph G such that Ḡ is C4-free and edges of Ḡ are all determined edges
On self-complementation
We prove that, with very few exceptions, every graph of order n, n - 0, 1(mod 4) and size at most n - 1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs
Hamilton cycles in 5-connected line graphs
A conjecture of Carsten Thomassen states that every 4-connected line graph is
hamiltonian. It is known that the conjecture is true for 7-connected line
graphs. We improve this by showing that any 5-connected line graph of minimum
degree at least 6 is hamiltonian. The result extends to claw-free graphs and to
Hamilton-connectedness
Groups whose character degree graph has diameter three
Let be a finite group, and let denote the \emph{prime
graph} built on the set of degrees of the irreducible complex characters of
. It is well known that, whenever is connected, the diameter
of is at most . In the present paper, we provide a
description of the finite solvable groups for which the diameter of this graph
attains the upper bound. This also enables us to confirm a couple of
conjectures proposed by M.L. Lewis
Closed Quasi-Fuchsian Surfaces In Hyperbolic Knot Complements
We show that every hyperbolic knot complement contains a closed
quasi-Fuchsian surface.Comment: 69 pages, 27 figures. Made small changes suggested by refere
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