# Bilinear and Trilinear Partitions of a Graph

Sampathkumar, E. and Latha, L. P. (1994) Bilinear and Trilinear Partitions of a Graph. Indian Journal of Pure & Applied Mathematics, 25 (8). pp. 843-850.

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## Abstract

Let G = (V, E) be a graph. The bilinear partition number chi(2L) (G) Of G is the minimum order of a partition L of V such that for any two sets V-i and V-j in L, the subgraph (V-i boolean OR V-j) induced by V-i boolean OR V-j is a linear forest. The trilinear partition number chi(3L) (G) Of G is the minimum order of a partition L of V such that for any three sets V-i, V-j and V-k in L, the subgraph (V-i boolean OR V-j boolean OR V-k) is a linear forest. Theorem 1 - If Delta greater than or equal to 3 is the maximum degree of G, then, 1 + Delta/2] less than or equal to chi(2L), with equality for trees. If G is a cactus, chi(2L) less than or equal to 2 + Delta 2]. Theorem 2 - If every block of G is complete, and omega(G) is the clique number, then chi(2L) (G) = max {omega(G), 1 + Delta/2]}. Theorem 3 - If G is K-3-free, then 1 + Delta less than or equal to chi(3L), With equality for trees. Further, if G is a cactus, chi(3L) less than or equal to 2 + Delta.

Item Type: Article E Mathematical Science > Mathematics Department of > Mathematics Users 23 not found. 03 May 2021 07:19 04 Feb 2023 06:04 http://eprints.uni-mysore.ac.in/id/eprint/16230