Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A digraph D = ( V , A ) is said to be m -colored if its arcs are colored with m colors. An m -colored digraph D has a k -colored kernel if there exists K ? V such that (i) for every v ? V ? K there exist a q -colored directed path, with q ? k , from v to a vertex of K , and (ii) for every pair { u , v } ? K every directed path from u to v uses at least k + 1 colors.Given an m -colored digraph D , the color-class digraph of D , denoted C ( D ) , is defined as follows: the vertices of C ( D ) are the m colors of D , and ( i , j ) is an arc of C ( D ) if and only if there exist two consecutive arcs in D , namely ( u , v ) and ( v , w ) , such that ( u , v ) has color i and ( v , w ) has color j .A digraph D is said to be cyclically k -partite if there is a partition { V i } i = 0 k - 1 of its vertices in independent sets such that every arc in D is either a loop or a V i V i + 1 -arc (taken the index modulo k ). In Galeana-Sanchez (2012) it was proved that given an m -colored digraph D , if C ( D ) is cyclically 2-partite then D has a kernel by monochromatic paths (that is a 1-colored kernel).   extend this work and prove the following: Let D be a strongly connected m -colored digraph D such that, for some k ? 1 , C ( D ) is a cyclically ( k + 1 ) -partite digraph, with partition { C i } i = 0 k . (i) If for some part C j , no vertex of C j has a loop, then D has a k -colored kernel. (ii) For each i , with 0 ? i ? k , let D i be the subgraph of D induced by the set of arcs with color in C i , and for each vertex x of D let N C + ( x ) and N C - ( x ) be the set of colors appearing in the ex-arcs and in-arcs of x , respectively. If for some subdigraph D j , for every vertex x of D j  have that N C + ( x ) ? N C - ( x ) , then D has a k -colored kernel.As a direct consequence  obtain a proof of Richardson's Theorem in the case D is strongly connected, and a proof of a classical result by M. Kwa?nik (see Kwa?nik (1983) ) on the existence of k -kernels (a k -kernel of a digraph D = ( V , A ) is a set S ? V such that for any v ? V ? S , d D ( v , S ) ? k - 1 and for every pair { u , v } ? S , d D ( u , v ) ? k ) that asserts that if D is a strongly connected digraph such that every directed cycle has length congruent with 0 modulo k , then D has a k -kernel.
['Hortensia Galeana-Sánchez', 'Juan José Montellano-Ballesteros']
I am writing a research paper on a idea Richardson's Theorem for k -colored kernels in strongly connected digraphs
[{"from": "human", "value": "I am writing a research paper on a idea Richardson's Theorem for k -colored kernels in strongly connected digraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A digraph D = ( V , A ) is said to be m -colored if its arcs are colored with m colors. An m -colored digraph D has a k -colored kernel if there exists K ? V such that (i) for every v ? V ? K there exist a q -colored directed path, with q ? k , from v to a vertex of K , and (ii) for every pair { u , v } ? K every directed path from u to v uses at least k + 1 colors.Given an m -colored digraph D , the color-class digraph of D , denoted C ( D ) , is defined as follows: the vertices of C ( D ) are the m colors of D , and ( i , j ) is an arc of C ( D ) if and only if there exist two consecutive arcs in D , namely ( u , v ) and ( v , w ) , such that ( u , v ) has color i and ( v , w ) has color j .A digraph D is said to be cyclically k -partite if there is a partition { V i } i = 0 k - 1 of its vertices in independent sets such that every arc in D is either a loop or a V i V i + 1 -arc (taken the index modulo k ). In Galeana-Sanchez (2012) it was proved that given an m -colored digraph D , if C ( D ) is cyclically 2-partite then D has a kernel by monochromatic paths (that is a 1-colored kernel).   extend this work and prove the following: Let D be a strongly connected m -colored digraph D such that, for some k ? 1 , C ( D ) is a cyclically ( k + 1 ) -partite digraph, with partition { C i } i = 0 k . (i) If for some part C j , no vertex of C j has a loop, then D has a k -colored kernel. (ii) For each i , with 0 ? i ? k , let D i be the subgraph of D induced by the set of arcs with color in C i , and for each vertex x of D let N C + ( x ) and N C - ( x ) be the set of colors appearing in the ex-arcs and in-arcs of x , respectively. If for some subdigraph D j , for every vertex x of D j  have that N C + ( x ) ? N C - ( x ) , then D has a k -colored kernel.As a direct consequence  obtain a proof of Richardson's Theorem in the case D is strongly connected, and a proof of a classical result by M. Kwa?nik (see Kwa?nik (1983) ) on the existence of k -kernels (a k -kernel of a digraph D = ( V , A ) is a set S ? V such that for any v ? V ? S , d D ( v , S ) ? k - 1 and for every pair { u , v } ? S , d D ( u , v ) ? k ) that asserts that if D is a strongly connected digraph such that every directed cycle has length congruent with 0 modulo k , then D has a k -kernel."}]