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We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then define correct contraction procedure to be executed to get largest possible complete graph from given connected graph. We finally give a characterization for k-chromatic graphs and use it to settle Hadwigers conjecture.
Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable graphs have
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical
We show that we cannot avoid the existence of at least one directed circuit of length less than or equal to (n/r) in a digraph on n vertices with out-degree greater than or equal to r. This is well-known Caccetta-Haggkvist problem.
We settle the Path Decomposition Conjecture (P.D.C.) due to Tibor Gallai for minimally connected graphs, i.e. trees. We use this validity for trees and settle the P. D. C. using induction on the number of edges for all connected graphs. We then obtai
Let $lambda(m)$ be the $m$th coefficient of a modular form $f(z)=sum_{mgeq 1} lambda(m)q^m$ of weight $kgeq 4$, let $p^n$ be a prime power, and let $varepsilon>0$ be a small number. An approximate of the Atkin-Serre conjecture on the lower bound of t