Girth tenacity of some cube-like networks

The girth vertex (resp. edge) tenacity g τ v ( G ) (resp. g τ e ( G ) ) of a non-acyclic simple graph G is defined to be the maximum number k such that the removal of any k vertices (resp. edges) of G does not change its girth. In this paper, we mainly investigated the girth tenacity of the exchanged hypercube E H ( s , t ) with s , t ≥ 1 , the ternary n -cube Q n 3 and the exchanged ternary n -cube E 3 C ( r , s , t ) with n = r + s + t and r , s , t ≥ 0 . We proved that ( i ) g τ v ( E H ( 1 , 1 ) ) = 0 , g τ v ( E H ( s , t ) ) = 2 ⌊ 2 t + 1 3 ⌋ − 1 for s = 1 and t ≥ 2 , g τ v ( E H ( s , t ) ) = 2 ⌊ 2 s + 1 3 ⌋ − 1 for t = 1 and s ≥ 2 , g τ v ( E H ( s , t ) ) = ⌊ 2 t + 1 3 ⌋ 2 s + ⌊ 2 s + 1 3 ⌋ 2 t − 1 for min { s , t } ≥ 2 , and ( i i ) g τ v ( Q n 3 ) = 3 n − 1 − 1 , g τ e ( Q n 3 ) = n 3 n − 1 − 1 , and ( i i i ) g τ v ( E 3 C ( r , s , t ) ) = 3 n − 1 − 1 , g τ e ( E 3 C ( r , s , t ) ) = ( n + 2 ) 3 n − 2 − 1 . Some results on the girth tenacity of the hypercube were also listed.