Theoretical analysis demonstrates an upper bound of (m+1)/6 for the isolation number of non-triangle cycles in graphs, extending prior bounds and characterizing extremal graph families.
Key Points
To determine a sharp edge-based upper bound for the isolation number of cycles of length at least four in connected graphs not isomorphic to a 4-cycle.
Analyzed the isolation number ι(G, C') representing the minimum vertex set whose closed neighborhood disrupts all non-triangle cycles in a connected graph G with m edges.
Identified and classified the extremal graph families that achieve equality for the derived upper bound.
Proved that ι(G, C') ≤ (m+1)/6 for any connected graph G with m edges, provided G is not a 4-cycle.
Generalized existing bounds for triangle-free graphs and recovered the inequality ι(G, {C₄}) ≤ (m+1)/6, confirmed to be attained by infinitely many non-isomorphic graphs.