Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In an undirected graph G, a subset C@?V(G) such that C is a dominating set of G, and each vertex in V(G) is dominated by a distinct subset of vertices from C, is called an identifying code of G. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph G, let @c^I^D(G) be the minimum cardinality of an identifying code in G. ,  show that for any connected identifiable triangle-free graph G on n vertices having maximum degree @D>=3, @c^I^D(G)@?n-n@D+o(@D). This bound is asymptotically tight up to constants due to various classes of graphs including (@D-1)-ary trees, which are known to have their minimum identifying code of size n-n@D-1+o(1).  also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant c such that the bound @c^I^D(G)@?n-n@D+c holds for any nontrivial connected identifiable graph G.
['Florent Foucaud', 'Ralf Klasing', 'Adrian Kosowski', 'André Raspaud']
I am writing a research paper on a idea On the size of identifying codes in triangle-free graphs
[{"from": "human", "value": "I am writing a research paper on a idea On the size of identifying codes in triangle-free graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In an undirected graph G, a subset C@?V(G) such that C is a dominating set of G, and each vertex in V(G) is dominated by a distinct subset of vertices from C, is called an identifying code of G. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph G, let @c^I^D(G) be the minimum cardinality of an identifying code in G. ,  show that for any connected identifiable triangle-free graph G on n vertices having maximum degree @D>=3, @c^I^D(G)@?n-n@D+o(@D). This bound is asymptotically tight up to constants due to various classes of graphs including (@D-1)-ary trees, which are known to have their minimum identifying code of size n-n@D-1+o(1).  also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant c such that the bound @c^I^D(G)@?n-n@D+c holds for any nontrivial connected identifiable graph G."}]