ON SMALLEST ORDER OF TRIANGLE-FREE GRAPHS WITH PRESCRIBED (3, k)-DEFECTIVE CHROMATIC NUMBER

Mudin Simanihuruk
2018 Far East Journal of Mathematical Sciences (FJMS)  
A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on vertices receiving the same colour is at most k. The k-defective chromatic number Xk(G) of a graph G is the least positive integer m for which G is (m,k)-colourable. Let f(m,k) be the smallest order of a triangle-free graph G such that X k (G) = m. In this paper we study the problem of determining f( m, 1). We show that f(3, 1) = 9 and characterize the corresponding
more » ... minimal graphs. For m ~ 4, we present lower and upper bounds for f(m,l).
doi:10.17654/ms103040717 fatcat:xpznpo5afrge7kb72pq3h7jvli