Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: An r -dynamic proper k -coloring of a graph G is a proper k -coloring of G such that every vertex in V ( G ) has neighbors in at least min { d ( v ) , r } different color classes. The r -dynamic chromatic number of a graph G , written ? r ( G ) , is the least k such that G has such a coloring. By a greedy coloring algorithm, ? r ( G ) ? r Δ ( G ) + 1 ;  prove that equality holds for Δ ( G ) 2 if and only if G is r -regular with diameter 2 and girth 5.  improve the bound to ? r ( G ) ? Δ ( G ) + 2 r - 2 when ? ( G ) 2 r ln n and ? r ( G ) ? Δ ( G ) + r when ? ( G ) r 2 ln n .In terms of the chromatic number,  prove ? r ( G ) ? r ? ( G ) when G is k -regular with k ? ( 3 + o ( 1 ) ) r ln r and show that ? r ( G ) may exceed r 1.377 ? ( G ) when k = r .  prove ? 2 ( G ) ? ? ( G ) + 2 when G has diameter 2, with equality only for complete bipartite graphs and the 5-cycle. Also, ? 2 ( G ) ? 3 ? ( G ) when G has diameter 3, which is sharp. Hover, ? 2 is unbounded on bipartite graphs with diameter 4, and ? 3 is unbounded on bipartite graphs with diameter 3 or 3-colorable graphs with diameter 2. Finally,  study ? r on grids and toroidal grids.
['Sogol Jahanbekam', 'Jaehoon Kim', 'Suil O', 'Douglas B. West']
I am writing a research paper on a idea On r -dynamic coloring of graphs
[{"from": "human", "value": "I am writing a research paper on a idea On r -dynamic coloring of graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: An r -dynamic proper k -coloring of a graph G is a proper k -coloring of G such that every vertex in V ( G ) has neighbors in at least min { d ( v ) , r } different color classes. The r -dynamic chromatic number of a graph G , written ? r ( G ) , is the least k such that G has such a coloring. By a greedy coloring algorithm, ? r ( G ) ? r \u0394 ( G ) + 1 ;  prove that equality holds for \u0394 ( G ) 2 if and only if G is r -regular with diameter 2 and girth 5.  improve the bound to ? r ( G ) ? \u0394 ( G ) + 2 r - 2 when ? ( G ) 2 r ln n and ? r ( G ) ? \u0394 ( G ) + r when ? ( G ) r 2 ln n .In terms of the chromatic number,  prove ? r ( G ) ? r ? ( G ) when G is k -regular with k ? ( 3 + o ( 1 ) ) r ln r and show that ? r ( G ) may exceed r 1.377 ? ( G ) when k = r .  prove ? 2 ( G ) ? ? ( G ) + 2 when G has diameter 2, with equality only for complete bipartite graphs and the 5-cycle. Also, ? 2 ( G ) ? 3 ? ( G ) when G has diameter 3, which is sharp. Hover, ? 2 is unbounded on bipartite graphs with diameter 4, and ? 3 is unbounded on bipartite graphs with diameter 3 or 3-colorable graphs with diameter 2. Finally,  study ? r on grids and toroidal grids."}]