On tree-connection of generalized line graphs

Abstract

The line graph \( L(G) \) of a nonempty graph \( G \) has the set of edges in \( G \) as its vertex set where two vertices of \( L(G) \) are adjacent if the corresponding edges of \( G \) are adjacent. Let \( k > 2 \) be an integer and let \( G \) be a graph containing k-paths (paths of order \( k \)). The k-path graph \( P_k(G) \) of \( G \) has the set of k-paths of \( G \) as its vertex set where two distinct vertices of \( P_k(G) \) are adjacent if the corresponding k-paths of \( G \) have a \( (k-1) \)-path in common. Thus, \( P_2(G) = L(G) \) and \( P_3(G) = L(L(G)) \). Hence, the k-path graph \( P_k(G) \) of a graph \( G \) is a generalization of the line graph \( L(G) \). Let \( G \) be a connected graph of order \( n > 3 \) and let \( k \) be an integer with \( 2 < k 2 \), then \( P_3(T) \) is \( k \)-tree-connected. This conjecture was verified for \( k = 2, 3 \). In this work, we show that if \( T \) is a tree of order at least 6 containing no vertices of degree 2, 3, 4, or 5, then \( P_3(T) \) is 4-tree-connected and so verify the conjecture for the case when \( k = 4 \).