k-Lucas numbers and associated bipartite graphs

For a positive integer k2, the k-Fibonacci sequence {g_n^(k)} is defined as: g₁^(k)==g_k?2^(k)=0, g_k?1^(k)=g_k^(k)=1 and for n>k2, g_n^(k)=g_n?1^(k)+g_n?2^(k)++g_n?k^(k). Moreover, the k-Lucas sequence {l_n^(k)} is defined as l_n^(k)=g_n?1^(k)+g_n+k?1^(k) for n1. In this paper, we consider the relationship between g_n^(k) and l_n^(k) and 1-factors of a bipartite graph.