Note on the Coalition Number of the <i>d</i>th Power of the <i>n</i>-Path

In a graph G=( V,E ) , two disjoint sets V 1 , V 2 ⊆V are said to form a coalition, if neither V 1 nor V 2 is a dominating set of G , but V 1 ∪ V 2 is a dominating set of G . The sets V 1 and V 2 forming a coalition are said to be coalition partners. A coalition partition, called a c-partition, is a vertex partition π={ V 1 , V 2 ,⋯, V k } such that each V i ∈π satisfies the following conditions: V i is a singleton dominating set of G , or V i is not a dominating set of G but has a coalition partner V j ∈π , a non-dominating set of G . The maximum order k of a c-partition of G is called the coalition number, denoted by C( G ) . In this paper, we study the coalition numbers of the dth power of the n-path P n d , get the exact values of C( P n d ) for enough large n , and also provide some bounds of C( P n d ) for the other cases. As a special case, we get the exact values of C( P n 2 ) except for n∈{ 11,12,⋯,20 } .