arXiv subjects
Publications and source records attributed to G. R. Vijayakumar.
We settle affirmatively a conjecture posed in [S. M. Hegde, Set colorings of graphs, European Journal of Combinatorics 30 (4) (2009), 986--995]: If some subsets of a set X are assigned injectively to all vertices of a complete bipartite graph G such that the collection of all sets, each of which is the symmetric difference of the sets assigned to the ends of some edge, is the set of all nonempty subsets of X, then G is a star.
A root system in which all roots have same norm is known as a simply laced root system. We present a simple method of classifying all simply laced root systems.
B. D. Acharya has conjectured that if $\bigl(A_i: i=1, 2, ..., 2^{|X|}-1\bigr)$ is a permutation of all nonempty subsets of a set $X$ with at least two elements such that for each even positive integer $j<2^{|X|}-1$, $A_{j-1}\triangle A_j\triangle A_{j+1}=\emptyset$, then $|X|=2$. In this article, we show that if the cardinality of a set $X$ is more than four, then a permutation as described above indeed exists.