k-Lucas numbers and associated bipartite graphs

AbstractFor a positive integer k⩾2, the k-Fibonacci sequence {gn(k)} is defined as: g1(k)=⋯=gk−2(k)=0, gk−1(k)=gk(k)=1 and for n>k⩾2, gn(k)=gn−1(k)+gn−2(k)+⋯+gn−k(k). Moreover, the k-Lucas sequence {ln(k)} is defined as ln(k)=gn−1(k)+gn+k−1(k) for n⩾1. In this paper, we consider the relationship between gn(k) and ln(k) and 1-factors of a bipartite graph