Extensions of edge-coloured digraphs

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1 Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, Extensions of edge-coloured digraphs HORTENSIA GALEANA-SÁNCHEZ Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria México, D.F MÉXICO ROCÍO ROJAS-MONROY Facultad de Ciencias UAEMex Instituto Literario No. 100, Centro 50000, Toluca, Edo. de México MÉXICO Abstract: A digraph D is said to be an m-coloured digraph, if its arcs are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N V (D) of vertices of D is said to be a kernel by monochromatic paths of the m-coloured digraph D, if it satisfies the two following properties : (1) N is independent by monochromatic paths; i.e. for any two different vertices x, y N, there is no monochromatic directed path between them, and (2) N is absorbent by monochromatic paths; i.e. for each vertex u V (D) N, there exists a uv-monochromatic directed path, for some v N. In this paper we present a method to construct a large variety of m-coloured digraphs with (resp. without a kernel) kernel by monochromatic paths; starting with a given m-coloured digraph D 0. A previous result is generalized. Key Words: kernel, kernel by monochromatic paths, m-coloured digraph 2000 Mathematic Subject Classification: 05C20 1 Introduction For general concepts we refer the reader to [1]. Let D be a digraph; V (D) and A(D) will denote the sets of vertices and arcs of D respectively. Let S 1, S 2 V (D), an arc (u 1, u 2 ) of D will be called an S 1 S 2 -arc whenever u 1 S 1 and u 2 S 2 ; D[S 1 ] will denote the subdigraph of D induced by S 1 ; S 1 V (D). A set I V (D) is independent if A(D[I]) =. A kernel N of D is an independent set of vertices such that for each z V (D) N there exists a zn-arc in D. A digraph D is called a kernel-perfect digraph when every induced subdigraph of D has a kernel. And D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. A digraph D is called an m-coloured digraph whenever its arcs are coloured with m colours. If D is an m-coloured digraph; the closure of D, denoted C(D) is the m-coloured multidigraph defined as follows: V (C(D)) = V (D) and (u, v) A(C(D)) with colour i if and only if, there exists an uv-monochromatic directed path in D, coloured i. Clearly D has a kernel by monochromatic paths if and only if C(D) has a kernel. And D has a kernel if and only if the m-coloured digraph D (where every two different arcs have different colours) has a kernel by monochromatic paths. In [4] was introduced the s-construction and was defined a digraph s(s) associated to a given digraph D 0 ; in the same paper was proved that s(s) has a kernel if and only if D 0 has a kernel. Also in [4] was proved that given a kernel-perfect digraph D 0, it is possible to construct a large variety of critical kernel imperfect digraphs containing D 0 as an induced subdigraph. More results concerning the existence of kernels in s- constructions and s-extensions of a given digraph D 0 can be found in [5] and [6]. Sufficient conditions for the existence of a kernel in a digraph have been investigated by several authors, namely Von Neumann and Morgenstern [13], Richardson [10], Duchet and Meyniel [2] and Galeana-Sánchez and Neumann-Lara [3]. The concept of a kernel is very useful in applications; and clearly the concept of a kernel by monochromatic paths generalizes those of a kernel. Sufficient conditions for the existence of a kernel by monochromatic paths in m-coloured digraphs have been investigated by several authors: for example in [12] Sands et al. proved that any 2-coloured digraph has a kernel by monochromatic paths; in [11] Shen Minggang proved that any m-coloured tournament in which each subtournament of order 3 is 2-coloured has a kernel by monochromatic paths. In [7] it was proved that if T is an m-coloured tournament such that every directed cycle of length at most 4 is quasi-monochromatic then

2 Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, C(T ) is kernel perfect. A generalization of this result was obtained by Hahn, Ille and Woodrow in [9] they proved that if T is an m-coloured tournament such that every directed cycle of length k is quasimonochromatic and T has no polychromatic directed cycles of length l, l < k, for some k 4; then T has a kernel by monochromatic paths. (A directed cycle is called quasi-monochromatic if with at most one exceptions all of its arcs are coloured alike, and a directed cycle is called polychromatic whenever its arcs allows at least tree colours Kernels by monochromatic paths in bipartite tournaments were studied in [8], where it is proved that if T is a bipartite tournament such that every directed cycle of length 4 is monochromatic, then T has a kernel by monochromatic paths. In this paper we define the s-construction which generalizes de s-construction. Also we define an m- coloured digraph s( S) related to a given m-coloured digraph D 0 ; and we prove that s( S) has a kernel by monochromatic paths if and only if D 0 has a kernel by monochromatic paths. This results generalizes the main result of [4] (s(s) has a kernel if and only if D 0 has a kernel). 2 Systems and Extensions In this section we introduce the s-construction. The main result concerning the s construction is Theorem 2.2 which enables us to generate a large class of m-coloured digraphs with (resp. without) kernel by monochromatic paths. Definition 2.1 Let D 0 be an m-coloured digraph. A 4 tuple S 0 = (C(D 0 ), U, U +, U ) will be called an s 0 -system (over D 0 ) if it satisfies: (i) U, U + and U are sets of vertices with the same cardinality and U V (D 0 ). (ii) V (D 0 ), U + and U are mutually disjoint sets. (iii) C(D 0 ) is the closure of D 0. (iv) If u U, then there is no monochromatic directed cycle contained in D 0, passing by u. Lemma 2.1 [1]. Let D 0 be a digraph and C a closed directed walk in D 0. Then for each u V (C), there exists a directed cycle contained in C, passing by u. Lemma 2.2 Let D be an m-coloured digraph, C(D) its closure and u V (D). There exists a monochromatic directed cycle, coloured i, contained in D and passing by u if and only if there exists a monochromatic directed cycle coloured i, contained in C(D) and passing through u. Proof: If C is a monochromatic directed cycle coloured i, contained in D, and passing by u. Then from the definition of C(D), clearly we have C C(D). Now suppose that there exists a monochromatic directed cycle, γ, coloured i, contained in C(D) and passing by u. Let γ = (u = z 0, z 1,..., z n = u); from the definition of C(D), we have that for each j 1, 2,..., n; there exists a z j 1 z j -monochromatic directed path coloured i, clearly the union of these paths is a closed monochromatic directed walk contained in D and passing through u. Thus from Lemma 2.1 there exists a directed cycle coloured i, contained in D and passing by u. In what follows, if S 0 = (C(D), U, U +, U ) is an s 0 -system, we shall denote by u + (resp.u ) the vertex in U + (resp. U ) which corresponds to u U for any fixed bijection from U to U + (resp. from U to U ). Definition 2.2 If S 0 = (C(D 0 ), U, U +, U )is an s 0 - system, we denote by s 0 ( S 0 ) the digraph defined as follows: V ( s 0 ( S 0 )) = (V (D 0 ) U) U + U ; (z, w) is an arc of s 0 ( S 0 ) coloured i if and only if one of the following conditions holds: (i) {z, w} V (D 0 ) U and there exists an arc from z to w in C(D 0 ), coloured i. (ii) z V (D 0 ) U, w = u for some u U; and there exists an arc from z to u in C(D), coloured i. (iii) z = u + for some u U, w V (D 0 ) U; and there exists an arc from u to w, in C(D), coloured i. (iv) z = u + and w = v, for some {u, v} U; and there exists an arc from u to v, in C(D), coloured i. Definition 2.3 An s-system is a 4-tuple S = ( S 0, β, Ũ+, Ũ ) where: (i) S 0 = (C(D 0 ), U, U +, U ) is an s 0 -system and Ũ + and Ũ are m-coloured multidigraphs such that V (Ũ+) = U + and V (Ũ ) = U. (ii) β = { β u u U} is a set of mutually disjoint directed paths where each β u is a u u + -directed path of positive even length and V (β u ) V ( s 0 ( S 0 )) = {u, u + }. (iii) for each u U; any two consecutive arcs of β u have different colours and the arc of β u which incides from u (resp. incides to u + ) has a colour which is different from the colour of each arc which incides to u (resp. from u) in D 0. Definition 2.4 Let D 0 be an m-coloured digraph. If S = ( S 0, β, Ũ+, Ũ ) is an s-system, we denote by s( S) the edge-coloured multidigraph defined as follows: s( S) = s 0 ( S 0 ) u U β u Ũ+ Ũ. The

3 Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, multidigraph s( S) will be called an extension of D 0. Notice that if U = then s( S) = s 0 ( S 0 ) = C(D 0 ). Definition 2.5 We will say that the multidigraph s( S) satisfies property (A) if the following condition holds: If there exists an arc from u + to v + (resp. from u to v ) in Ũ+ (resp. in Ũ ) coloured i, then there exists an arc from u to v in C(D 0 ), coloured i. Definition 2.6 We will denote by g the function defined as follows: g : V ( s 0 ( S 0 )) V (D 0 ); g(z) = { u if z = u or z = u + z if z U + U. Theorem 2.1 Let S = ( S0, β, Ũ+, Ũ ) be an ssystem where S 0 = (C(D 0 ), U, U +, U ). Suppose that s( S) satisfies property (A). Then the following conditions hold. (1) Each U U + -directed path contained in s( S) contains a directed path β u, for some u U. (2) If T is a monochromatic directed path contained in s( S) of length at least two, then V (T ) ( u U V ( β u ) (U + U )) =. (3) For any u U; there is no monochromatic directed path between u + and u contained in s( S). (4) If there exists a zw-monochromatic directed path coloured i, contained in s( S) and with length at least two then {z, w} V ( s 0 ( S 0 )), and there exists a g(z)g(w)-directed path coloured i (monochromatic) contained in D 0. (5) If T is a u z-monochromatic directed path contained in s( S) and β u = (u = z 0, z 1,..., z n = u + ), then z = z 1 or V (T ) U. (6) If T is a zw-directed path coloured i, contained in s( S) such that z V ( s 0 ( S 0 )) U. Then V (T ) ( u U V ( β u ) (U + U )) =, and there exists a g(z)g(w)-directed path coloured i, contained in D 0. Proof: (1). Let T = (v = z 0, z 1,..., z n = v ) a vv -directed path with v U and v U +. Denote by j 0 = min{j {0, 1,..., n} z j+1 U } and by j 1 = min{j {0, 1,..., n} j > j 0 and z j U + }; thus z j0 U, z j0+1 U, z j1 U +. Now we consider T 0 = (z j0, T, z j1 ) (the z j0 z j1 -directed path contained in T ). Let u U be such that z j0 = u, from the definition of s( S) we have that every arc of s( S) starting in u ends in U or in the second vertex of β u ; since z j0+1 U we have z j0+1 β u. We conclude that β u = T 0. (as z ji U + we have z ji = u + ). (2) Suppose that T is a zw-directed path coloured i, contained in s( S) with length at least two. Let u U be and x V ( β u ) {u +, u }. Since the arcs incident with x belong to β u ; and thus they have different colours; we obtain x (V (T ) {z, w}). Now; if x = z, then the next vertex of T must be u + ; and from definition of s-system, the colour of (x, u + ) is different from the colour of the next arc of T ; a contradiction (as T is monochromatic). If x = w,then (u, x) A(T ). From the definition of s-system we have that (u, x) has a colour which is different of the colour of any arc incident toward u ; so T is not monochromatic; a contradiction. (3) Assume by contradiction that there exists a monochromatic directed path, between u + and u, for some u U,contained in s( S); and let T 0 be such a directed path of minimum length. Let û U be such that û + and û are the terminals of T 0. Since T 0 is monochromatic, coloured, say i; it follows from the definition of s( S) that l(t 0 ) 2. Therefore from (2) we have: V (T 0 ) ( u U V ( β u ) (U + U )) = ; the function g is defined on V (T 0 ); and T 0 contains no β u for every u U. Thus from (1) we have that T 0 is a û + û -directed path coloured i. From the choice of T 0, we have that for any u U {û}, {u +, u } V (T 0 ). Thus, function g restricted to V (T 0 ) {û } is an injective function. If T 0 = (û + = z 0, z 1,..., z n = û ) then it follows from the definition of s 0 ( S 0 ) that for each j {1,..., n} there exists an arc from g(z j 1 ) to g(z j ) coloured i, contained in C(D 0 ). Therefore C = (u = g(z 0 ), g(z 1 ),..., g(z n ) = u) is a monochromatic directed cycle, contained in C(D 0 ), an passing through u; contradicting that S 0 is an s 0 - system. (4) Let T be a zw-directed path, coloured i, contained in s( S), and with l(t ) 2. From (2) we have V (T ) ( u U V ( β u ) (U + U )) = ; thus V (T ) V ( s 0 ( S 0 )) and g is defined for every vertex of T. Now; if T = (z = z 0, z 1,..., z n = w) then from the definition of s 0 ( S 0 ) we have that for each j {1,..., n} there exists in C(D 0 ), an arc coloured i, from g(z j 1 ) to g(z j ); therefore there exists a g(z j 1 )g(z j )- directed path coloured i contained in D 0. We conclude that there exists a g(z)g(w)- directed path coloured i, contained in D 0. (5) Let T be a u z- monochromatic directed path, contained in s( S) and assume that βu = (u = z 0, z 1,..., z n = u + ). Assume by contradiction that T (u, z 1 ) and V (T ) U. Let T = (u = w 0, w 1,..., w m = z). First observe that w 1 U (otherwise, from the defintion of s( S) we have w 1 = z 1 ; and the coloring of the arcs of β u implies T = (u, z 1 ), contradicting our assumption). Denote by

4 Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, j 0 = min{j {0, 1,..., m} w j+1 U }. Thus, j 0 1, w k U for each k, 0 k j 0 and w j0+1 U. Let v U be such that w j0 = v, then w j0+1 is the second vertex of β v. Now from the definition of s-system, we have that the arcs (w j0 1, w j0 = v ) and (w j0 = v, w j0+1 ) have different colours; contradicting that T is monochromatic. (6) Let T be a zw-directed path, coloured i, contained in s( S) and such that z (V ( s 0 ( S 0 )) U ). If l(t ) 2, then the assertion follows from (2) and (4). If T = (z, w), then w u U (V ( β u {u }) (as z (V ( s 0 ( S 0 )) U ) and w ( u U V ( β u ) (U + U )). So V (T ) ( u U V ( β u ) (U + U )) =. It follows from that function g is defined for z and w; and from the definition of s( S), there exists an arc coloured i from g(z) to g(w) in C(D 0 ) i.e. there exists a g(z)g(w)-directed path coloured i in D 0. Theorem 2.2 Let S = ( S0, β, Ũ+, Ũ ) be an ssystem where S 0 = (C(D 0 ), U, U +, U ). Suppose that s( S) satisfies property (A). Then s( S) has a kernel by monochromatic paths if and only if D 0 has a kernel by monochromatic paths. Proof: Suppose D 0 has a kernel by monochromatic paths N 0, and U. For each u U let N u be defined as follows: if u N 0, then N u is a kernel by monochromatic paths of β u and if u N 0, then N u is a kernel by monochromatic paths of β u {u + }. Clearly from the definition of s-system we have the following two assertions. 1. The kernel by monochromatic paths of β u contains {u +, u } and 2. The kernel by monochromatic paths of β u {u + } does not contain u. We will prove that N = (N 0 U) u U N u is a kernel by monochromatic paths of s( S). 3. For each u U, u N 0 if and only if {u, u + } N. First we suppose u N 0. From the definition of N u, we have that N u is a kernel by monochromatic paths of β u, from 1 we have {u, u + } N u and from the definition of N; N u N. Now suppose {u, u + } N; since the directed paths β u are mutually disjoint, we have {u, u + } N u ; moreover from 1 and 2 we have N u is a kernel by monochromatic paths of β u ; and finally from the definition of N u we have u N N is independent by monochromatic paths in s( S). Let {z, w} N and assume by contradiction that there exists a zw-monochromatic directed path (say T ) in s( S). Case 4(a). {z, w} V ( s 0 ( S 0 )). From (4) in Theorem 2.1 and the definitions of s( S) and g; we have that there exists in D 0 a g(z)g(w)-monochromatic directed path when l(t ) 2. When l(t ) = 1 it follows from condition (A) that there exists a g(z)g(w)-monochromatic directed path in D 0. Since {z, w} V ( s 0 ( S 0 )), then {z, w} (N 0 U) (( u U N u) (U + U )). And we have three subcases: Case 4(a.1). {z, w} N 0 U. In this case g(z) = z,g(w) = w. And there exists a zw-monochromatic directed path in D 0 ; a contradiction (as N 0 is a kernel by monochromatic paths of D 0 ). Case 4(a.2). z (N 0 U) and w (( u U N u ) (U + U )) (analogously z (( u U N u) (U + U )) and w (N 0 U). Now w {u +, u } for some u U and w N u. From 1 and 2, {u, u + } N u, {u, u + } N and from (3) u N 0.Therefore there exists a zumonochromatic directed path (g(z) = z, g(w) = u) contained in D 0 a contradiction. Case 4(a.3). {z, w} (( u U N u) (U + U )). g(z) = u and g(w) = v for some {u, v} N 0 U. If u = v, then {z, w} = {u +, u } and there exists a monochromatic directed path in s( S) between u and u +, contradicting (3) in Theorem 2.1. If u v, then there exists a uv-monochromatic directed path contained in D 0 and with {u, v} N 0 ; a contradiction. Case 4(b). {z, w} V ( s 0 ( S 0 )) =. In this case {z, w} ( u U N u) (U + U ). Since each N u is a kernel by monochromatic paths; if follows that z N u and w N v, for some u, v U, u v.thus z V ( β u ) and w V ( β v ). Clearly; z {u +, u } from the definition of s( S) we have l(t ) = 1, moreover since the only arc which incides from z is in β u we have w V ( β u ), contradicting that V ( β u ) V ( β v ) =. Case 4(c). z V ( s 0 ( S 0 )) and w V ( s 0 ( S 0 )) (analogously the case z V ( s 0 ( S 0 )) and w V ( s 0 ( S 0 ))). In this case z N u {u +, u } for some u U and w N 0 U. From the definition of s( S) there is no monochromatic directed path between z and w in s( S); a contradiction. 5. N is absorbent by monochromatic paths. Let x (V ( s( S)) N); we will prove that there exists a xz-monochromatic directed path, for some z N. Case 5(a). x V ( s 0 ( S 0 )) (U + U ). In this case g(x) = x and x (V (D 0 ) N 0 );

5 Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, then there exists an xy-monochromatic directed path, for some y N 0. Thus there exists an arc from x to y in C(D 0 ). When y U, clearly we have in s( S) an arc from x to y with y N. When y U, then there exists an arc from x to y in s( S), and since y N 0, we have y N. Case 5(b). x V ( β u ) {u +, u } for some u U. Since x N,then x N u ; and there exists an xz-monochromatic directed path for some z N u (as N u is a kernel by monochromatic paths of β u or of ( β u {u + }). Case 5(c). x U + i.e. x = u + for some u U. Since x N; then we have u N 0. From the definiton of N 0 ; there exists an uy-monochromatic directed path in D 0, for some y N 0. Thus, there exists an arc from u to y in C(D 0 ).When y U; we have that there exists in s( S) an arc from u + to y, and from the definition of N; y N. When y U we obtain that there exists an arc from u + to y in s( S), and y N (as y N 0 ). Case 5(d). x U, i.e. x = u, for some u U. Since x N, then from the definition of N, we have x N u i.e. u N u. Therefore N u is not a kernel by monochromatic paths of β u and then N u is a kernel by monochromatic paths of β u {u + }. Thus, there exists a u z-monochromatic directed path in β u {u + }, with z N u N. We conclude from 4 and 5 that N is a kernel by monochromatic paths of s( S). Now suppose that N is a kernel by monochromatic paths of s( S). First we prove the following assertion: 6. u + N if and only if u N. First suppose u + N and let βu = (u = z 0, z 1,..., z n = u + ), it follows from the definition of s-system that z 1 N. Now assume by contradiction u N. Since N is a kernel by monochromatic paths of s( S), then there exists a u z-monochromatic directed path, say, T, in s( S) for some z N; since z 1 N it follows from (5) Theorem 2.1 that V (T ) U. Therefore the function g restricted to V (T ) is injective and from the condition (A) we have that there exists a g(u )g(z)-monochromatic directed path contained in C(D 0 ). It follows that there exists an arc from g(u ) to g(z) in C(D 0 ), and from the definition of s( S), there exists an arc from u + to û in s( S) with û U, z = û ; i.e. there exists an arc from u + to z in s( S) with {u +, z} N; a contradiction. Now suppose u N; and let βu = (u = z 0, z 1,..., z n = u + ). Since N is independent by monochromatic paths, we have z 1 N. Now z 2 N (from the definition of s-system), and z 3 N; continuing this way, we get u + N. Now we will prove that N 0 = {g(z) z (N ( u U V ( β u ) (U + U )))} is a kernel by monochromatic paths of D N 0 is independent by monochromatic paths in D 0. Let z, w (N ( β u U u ) (U + U )) be such that g(z) = x and g(w) = y. Assume by contradiction that there exists an xy-monochromatic directed path in D 0 ; this implies that there exists an arc from x to y in C(D 0 ). We will analyze the following four possible cases: 7(a). x U and y U. In this case z = x, w = y. From definiton of s( S); there exists an arc from z to w in s( S) with {z, w} N; a contradiction. 7(b). x U and y U. Now, w = y and we may assume (from 6) that z = x +. From definition of s( S); there exists an arc from x + to y in s( S) i.e. from z to w with {z, w} N; a contradiction. 7(c). x U, y U. In this case z = x and from 6 we may assume w = y. From definition of s( S) there exists an arc from x to y in s( S) i.e. an arc from z to w with {z, w} N; a contradiction. 7(d). x U, y U. From 6 we may assume z = x + and w = y. From definition of s( S) there exists an arc from x + to y in s( S) i.e. an arc from z to w with {z, w} N; a contradiction. 8. N 0 is absorbent by monochromatic paths in D 0. Let x (V (D 0 ) N 0 ) be; we will prove that there exists an xy-monochromatic directed path in D 0, for some y N 0. 8(a). x U. In this case x (V ( s 0 ( S 0 )) (U + U )) and then g(x) = x. Since x N 0, then x N; and there exists an xw-monochromatic directed path, say, T, in s( S), for some w N. Now from (6) in Theorem 2.1 we have V (T ) ( u U V ( β u ) (U + U )) = and there exists a g(x)g(w)-monochromatic directed path in D 0.Thus g(w) N 0 (recall w N), and there exists an xy-monochromatic directed path in D 0, with y N 0, y = g(w). 8(b). x U. Since x N 0, then x + N and x N. Therefore there exists an x + w-monochromatic directed path in s( S), say T, for some w N. From (6) in Theorem 2.1, we have V (T ) ( u U V ( β u ) (U + U )) = and there exists a g(x + )g(w)- monochromatic directed path in D 0, ( as x + U ); moreover g(w) N 0 (because w N). We conclude

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