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Prof. Suresh M Hegde
National Institute of Technology Karnataka, India
A Partial Solution of the Erdös - Faber - Lovász Conjecture

In 1972, Erdös - Faber - Lovász (EFL) conjectured that, if H is a linear hypergraph consisting of n edges of cardinality n, then it is possible to color the vertices with n colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erdös and Frankl had given an equivalent version of the same for graphs: Let G=i=1Ai denote a graph with n complete graphs A1,A2,… ,An, each having exactly n vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of G is n. 
The clique degree d^K (v) of a vertex v in G is given by d^K (v) = |{A_i: v∈V(A_i),1≤ i≤ n}|. The maximum clique degree Δ^K (G) of the graph G is given by Δ^K (G)=max_(v∈V(G) ) d^K (v). In this paper, using Symmetric Latin Squares, we give a constructive proof of the above conjecture for all graphs G in which the clique degree of every vertex, d^K (v) is either 1or m, where 3≤m≤n. 

Prof. Suresh M Hegde received his M.Sc. in Mathematics from Karnataka University Dharwad & Ph.D. from the Delhi University, New Delhi, India. He is currently a Professor in the Mathematical & Computational Sciences department, National Institute of Technology Karnataka, INDIA. Prof. Suresh M Hegde has worked as the Department Chair and in other capacities in the institute. Presently working as the Coordinator for Center for Continuing Education in the institute. Prof. Hegde visited many departments/universities like, Department of Applied Mathematics and Computer Science, Chulalongkorn Bangkok, Dept. of Mathematics, Hongkong University of Science and Technology, Dept. of Applied Mathematics, National Chaio Tung University, Taiwan, Dept. of Applied Mathematics, Thunghai University, Dept. of Mathematics, University of Alabama, Dept. of Mathematics, Purdue University, Indiana, Dept. of Electrical Engineering and Computer Science, University of Central Florida, during Dept. of Electrical Engineering and Computer Science, University of Newcastle, Australia, etc, for collaborative research. Prof. Hegde’s research interests include: Graph labelings, graph colorings, graph operators. He has guided/guiding Ph. D students and published extensively in these areas. (8 students and 50 journal papers). He has given invited talks at the National and International Conferences held in India and Abroad. Prof. Hegde has chaired international conferences and served in international conference committees.
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