ON THE NUMBER OF ALTERNATING PATHS IN BIPARTITE COMPLETE GRAPHS

ON THE NUMBER OF ALTERNATING PATHS IN BIPARTITE COMPLETE GRAPHS PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE December 1, 016 Abstract. Let C [r] m be a code such that any two words of C have Hamming distance at least t. It is not difficult to see that determining a code C with the maximum number of words is equivalent to finding the largest n such that there is an r-edge-coloring of K m,n with the property that any pair of vertices in the class of size n has at least t alternating paths with adjacent edges having different colors) of length. In this paper we consider a more general problem from a slightly different direction. We are interested in finding maximum t such that there is an r-edge-coloring of K m,n such that any pair of vertices in class of size n is connected by t internally disjoint and alternating paths of length k. We also study a related problem in which we drop the assumption that paths are internally disjoint. Finally, we introduce a new concept, which we call alternating connectivity. Our proofs make use of random colorings combined with some integer programs. 1. Introduction In this paper we study alternating paths in bipartite graphs. A path is alternating if its adjacent edges have different colors. The notion of alternating paths was originally introduced by Bollobás and Erdős [4], where the authors studied under which conditions an r-edge-colored complete graph contains an alternating Hamiltonian cycle. There is a broad literature on this subject for graphs, see, e.g., [1,, 5, 11], and also for hypergraphs [6, 7]. Several other results, mainly of algorithmic nature, are also known, see, e.g., a survey paper [3]. The motivation of this paper comes from coding theory. Recall that the Hamming distance between vectors x and y in [r] m is defined to be the number of positions in which they differ, i.e., distx, y) = {i : 1 i m, xi) yi)}. Let α r m, t) be the maximum size of a code C [r] m such that any two elements of C have Hamming distance at least t. We refer the reader to [9] for more details concerning coding theory. Let K m,n be a complete bipartite graph on vertex set [m] [n]. Suppose that c : EK m,n ) [r] is an r-edge-coloring of K m,n with the property that every pair of vertices in [n] is connected by at least t alternating paths of length with 3 vertices). The second author was supported in part by Simons Foundation Grant #4471 and by the National Security Agency under Grant Number H9830-15-1-017. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon. 1

PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE This edge coloring can be represented as a collection of n vectors of length m with entries in [r] in the sense that for a vertex v [n] we define the vector c v by c v u) = c{v, u}) for u [m]. Hence the set of vectors C = {c v : v [n]} completely encodes the edge coloring since every edge of K m,n will belong to exactly one of these vectors, and by looking at the right one we can determine the color assigned to the edge. Now notice that the number of alternating paths of length between v, w [n] is exactly the Hamming distance distc v, c w ). This is because a path of length is of the form v u w which is alternating if and only if c{v, u}) c{u, w}) or equivalently c v u) c w u). Since c has the property that every pair of vertices in the same partite set is connected by at least t alternating paths of length it follows that C has minimum Hamming distance t, thus we must have that C = n α r m, t). Consequently, determining α r m, t) is equivalent to finding the largest n such that there is an r-coloring of the edge set of K m,n with the property that any pair of vertices in [n] has at least t alternating paths of length connecting them. Clearly all such paths are internally disjoint. In this paper, we approach the problem from a slightly different direction: instead of fixing the alphabet, word length, and t and asking for the largest possible code, we fix the alphabet, word length, and code size and ask for the largest possible t. We also consider longer paths. Let κ r,k m, n) be the maximum t such that there is an r-coloring of the edges of K m,n such that any pair of vertices in class of size n is connected by t internally disjoint and alternating paths of length k. As noted above, κ r, m, n) is related to coding theory. However we study κ r,k m, n) for general k. In terms of coding theory, each such path is an alternating sequence of codewords and indices, such that each pair of consecutive codewords use different letters in the index that connects them in the sequence. We will show that for r and n m log n and for any k and n m 1 κ r, m, n) 1 1 r κ r,k m, n) m k. ) m, These results are essentially best possible since for m < log r n we have κ r, m, n) = 0. Indeed, if m < log r n, then n > r m and so there must be two vertices in the class of size n that have the same vectors of colors, and so they have no alternating path of length connecting them. We will also consider a related problem in which we drop the requirement that paths are internally disjoint. Let λ l m, n) be the maximum t such that there is a -coloring of the edges of K m,n such that any pair of vertices is connected by t alternating paths of length l. If l is even, then we consider only pairs in the partition class of size n. Determining λ l m, n) seems to be more difficult. In particular, we will show that λ 3 m, n) mn/4 and λ 4 m, n) m n/8. In doing so we determine an optimal upper bound on the number of alternating paths of length 3 and 4 in K m,n under any -edge-coloring.

Motivated by studying κ r,l m, n), in the last section, we propose a new concept, which we call alternating connectivity and define it as maximum t such that there is an r-edge-coloring of G such that any pair of vertices is connected by t internally disjoint and alternating paths of length l. We will discuss it briefly and show that for complete graphs κ r, K n ) 1 1/r)n and κ r,l K n ) n/l 1) for any r and l 3. Throughout this paper all asymptotics are taken in n. For simplicity, we do not round numbers that are supposed to be integers either up or down; this is justified since these rounding errors are negligible to the asymptomatic calculations we will make. All logarithms are natural unless written explicitly.. Paths of length In this section we only consider paths of length. As it was already mentioned in the introduction here instead of fixing the alphabet [r], word length m, and t and asking for the largest possible code in [r] m with the minimum Hamming distance t, we fix the alphabet, word length, and code size and ask for the largest possible t. Theorem.1. Let r and n m log n. Then, κ r, m, n) 1 1 ) m. r This is essentially the best possible since, as it was observed in the introduction, if m < log r n), then κ r, m, n) = 0. The proof of this theorem is an immediate consequence of Lemmas. and.3. Lemma.. Let r and n m be positive integers. Then, κ r, m, n) 1 1 ) 1 + 1 ) m 1 1 ) m. 1) r n 1 r Proof. Let K m,n be an r-edge-colored bipartite graph on [m] [n]. For a vertex v [m] let deg i v) denote the number of edges adjacent to v which are colored by i. Note that the total number of alternating paths of length with the middle vertex v is 1 i<j r deg iv) deg j v), which is r deg i v) deg j v) = 1 r deg i v)) deg i v) 1 i<j r = 1 i=1 n i=1 ) r deg i v) 1 i=1 ) n n = r 1 r r n, where the last inequality follows from the Cauchy-Schwarz inequality. Thus, ) n κ r, m, n) m r 1 r n, which is equivalent to 1). 3

4 PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE Lemma.3. Let r and n m log n. Then, κ r, m, n) 1 1 ) m om). r Proof. Let [m] [n] be the set of vertices of K = K m,n. To each edge in K we assign a color from {1,..., r} uniformly at random. For u, v [n], let X u,v be the random variable that counts the number of alternating paths between u and v of length. Clearly, X u,v Binm, r 1)/r). We will use the following form of Chernoff s bound see, e.g., Theorem.1 in [8]) PrZ EZ) t) exp t EZ) where Z is a random variable with binomial distribution. Since EX u,v ) = r 1)m/r and m log n, we get for t = 5EX u,v ) log n that EX) t = r 1)m/r om) and so Pr X u,v r 1 m om) r ), ) ) t exp = exp 5log n)/). EX u,v ) Thus, the union bound taken over all n ) exp log n) pairs of vertices in [n] yields the statement. 3. Paths of arbitrary length Here we consider κ for paths of any length. Quite surprisingly two colors already suffice to get an optimal result. Theorem 3.1. Let r, k and n m 1. Then, κ r,k m, n) m k. The only case not covered by the above theorem is when m is a constant. For example, in this regime it is not difficult to see that κ,4, n) = 0 and κ,4 3, n) = 1 for any n sufficiently large. The proof of Theorem 3.1 is based on the following lemma. Denote by Gm, m, 1/) the truly) random bipartite graph in which every possible edge between two partition classes of size m is chosen independently with probability 1/. We say that an event E n occurs with high probability, or w.h.p. for brevity, if lim n PrE n ) = 1. Lemma 3.. Let m 1 and let 0 < α < 1. Let Gm, m, 1/) be a random bipartite graph on set M 1 M with M 1 = M = m. Then, w.h.p. for each A M 1 αm) and B M αm) there exists a matching between A and B of size αm om). Proof. Fix A M 1 ) αm and B M αm) and let G = G A, B, 1/). First we consider an auxiliary bipartite graph H on U W such that U = A A, W = B B, H[A B] = G, and H[A W ] and H[U B ] are complete bipartite graphs. Furthermore, let s = log m = A = B. We show that H has a perfect matching. It suffices to show that the Hall condition holds, i.e., if S U and S U /, then NS) S, )

and if T W and T W /, then NT ) T. 3) If S < s, then since NS) B, NS) B = s S. Therefore, we assume that s S U /. Furthermore, we may assume that S A =. For otherwise, NS) = W. We will show that already for S = s, NS) W / = U /. Suppose not, that is, NS) < αm + log m)/. That means B NS) < αm log m)/, es, B \ NS)) = 0 and B \ NS) αm + log m)/ > αm/. Observe that the probability that there are sets S A and T B such that S = s and T = αm/ and es, T ) = 0 is at most ) ) αm αm s αm/ αm αm s αm/ = αm s αm/. s αm/ Thus, with probability at most αm s αm/ graph H violates ), and similarly 3). In other words, with probability at least 1 αm s αm/ graph H has a perfect matching, and consequently, there is a matching of size αm s between A and B. Finally, by taking the union bound over all A M 1 ) αm and B M αm) we get that the probability that there exist A and B such that between A and B there is no matching of size αm s is at most ) ) m m αm s αm/+1 m m αm s αm/+1 = o1), αm αm since s = log m. Also, clearly, we get that αm s = αm om). Thus, w.h.p. for each A and B there is a matching between A and B of size αm om). Proof of Theorem 3.1. First observe that the upper bound is trivial since 1+o1))m/k paths of length k saturate all vertices in M. Let M N be the bipartition of K = K m,n such that M = m and N = n. Furthermore, let N = N N, where N = m and N = n m. To each edge in K induced by M N we assign a color from {blue, red} uniformly at random. To all other edges between M and N ) we assign colors in such a way that for each v N, deg B v) = deg R v) and for each w M and u, v N, {u, w} and {v, w} have the same color in other words the color vector of each v N is the same). Observe that both the red graph induced on M N and the blue one can be viewed as Gm, m, 1/). So actually our r-edge-coloring uses only colors. First we consider u and v in N. Let X B u) and X R v) be two disjoint subsets of M such that all edges between u and X B u) are blue and all edges between v and X R v) are red and X B u) = X R v) = m/k. Now we choose subsets X 1,..., X k M and Y 1, Y,..., Y k 1 N which are pairwise disjoint and also disjoint with X B u) and X R v), and X i = Y j = m/k. By Lemma 3. there is a red matching between X B u) and Y 1, a blue matching between Y 1 and X 1, etc., each of size m/k om). This yields m/k om) alternating and internally disjoint paths between u and v. Now consider u and v in N and define N xy u, v) = {w M : {w, u} has color x and {w, v} has color y}. Chernoff s bound implies that N RR u, v) N RB u, v) N BR u, v) N BB u, v) m/4 for all u, v N. Let X B u) N BR u, v), X R v) N RR u, v) such that X B u) = 5

6 PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE X R v) = m/k) + om). In other words, all edges between u and X B u) are blue and all edges between v and X R v) are red. Now we choose disjoint subsets X 1,..., X k N BR u, v) N RR u, v) \ X B u) X R v)) and Y 1,..., Y k 1 N such that X i = Y j = m/k) + om). By Lemma 3. there is a red matching between X B u) and Y 1, a blue matching between Y 1 and X 1, etc., each of size m/k) om). This yields m/k) om) alternating and internally disjoint paths between u and v. To find the remaining m/k) om) paths we choose X R u) N RBu, v), X B v) N BBu, v), v)), and Y 1,..., Y X 1,..., X k N RBu, v) N BB u, v) \ X R u) X B Y k 1 ). Finally observe that the case u N and v N is very similar to the latter. k 1 N \ Y 1 Note that the above proof would not work for all m 1 if we used the simpler strategy of just coloring all edges randomly. In particular, to get the concentration of degrees we would need m to be at least on the order of log n. In a similar manner one can define κ r,k+1 m, n). Now, clearly the endpoints are in different partition classes. This case is not interesting since one can easily see that κ r,k+1 m, n) m/k. For the lower bound color by red one matching saturating all vertices in the class of size m and all other edges by blue. As in the previous proof m/k is best possible. 4. Not necessary internally disjoint paths In this section we are not assuming anymore that paths must be internally disjoint. Furthermore, we will only consider -coloring. Recall that λ l m, n) denotes the maximum t such that there is a -edge-coloring of K m,n such that any pair of vertices is connected by t alternating paths of length l. If l is even, then we consider only pairs in partition class of size n. In general determining λ seems to be more difficult than κ and as we will see the corresponding results for λ significantly differ from those for κ. We start with a lower bound. Theorem 4.1. Let k 1 be an integer. i) If n m log n, then ii) If n m = Ω1), then λ k m, n) 1 + o1))m k n k 1 / k 1. λ k+1 m, n) 1 + o1))mn/4) k. Proof. For ii) it suffices to consider the following coloring. Let M = M 1 M and N = N 1 N, M i = m/ and N i = n/, be partition classes of K m,n. Color the edges between M i and N i red and blue otherwise. It is easy to see that this coloring yields λ k+1 m, n) 1 + o1))m/) k n/) k. Now we show i). We use the concentration of degrees and codegrees. We fix vertices u and v in N. We then choose some x 1 M such that {u, x 1 } is red, some x N such that {x 1, x } is blue, and so on until we reach x k N. So far there are asymptotically m choices for each x i if i is odd, and n choices if i is even. Now we have to choose x k 1 such that the edge {x k, x k 1 } is red and {x k 1, v} is blue. There are asymptotically

m choices for x 4 k 1. This gives m k n k 1 / k choices for paths with a red edge adjacent to u. Similarly, we estimate the number of paths with a blue edge adjacent to u. It is not clear whether Theorem 4.1 is optimal in general. Here we managed to show tight upper bounds on λ 3 m, n) and λ 4 m, n). Clearly also λ m, n) = κ, m, n) m/. Theorem 4.. Let n m 1. Then, λ 3 m, n) mn/4 and λ 4 m, n) 1 + o1))m n/8. This theorem will immediately follow from the following result. Lemma 4.3. Let the edges of K m,n be -colored. Then, the number of all alternating paths of length 3 is at most m n /4 and the number of all alternating paths of length 4 with two endpoints in the class of size n is at most m n 3 /16. Indeed, this lemma implies that and λ 3 m, n) m n /4) / mn = mn/4, λ 4 m, n) m n 3 /16) / ) n = 1 + o1))m n/8. Proof of Lemma 4.3. Let K = K m,n be a blue-red edge-colored bipartite graph on [m] [n]. First we count the total number of alternating paths of length 3. Each such path has two vertices u, v [m] and two vertices x 1, x [n]. We assume that u and v are fixed and u < v. First we determine the number of red-blue-red paths containing u and v. Here we have two possibilities: either u x 1 v x is red-blue-red or x 1 u x v is red-blue-red. This yields codeg RB u, v) deg R v) + deg R u)codeg BR u, v) number of choices, where codeg xy u, v) = N xy u, v). Similarly, we see that the number of blue-red-blue paths containing u and v is codeg BR u, v) deg B v) + deg B u)codeg RB u, v). Thus, the number of alternating path of length 3 is never bigger than the solution of the following integer program: Maximize xbr u, v)[x R u) + x B v)] + x RB u, v)[x B u) + x R v)] ) subject to: The solution is graphical. We say that a solution is graphical if there exists a -edge-coloring of K m,n which realizes all color degrees and codegrees corresponding to the variables of the program. We will find an upper bound on the solution of this program. First we show that x BR u, v) x RB u, v) = x B u) x B v). 4) 7

8 PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE Let X B u) and X B v) be the set of vertices that are connected to u and v by blue edges, respectively. Furthermore, let X BR u, v) be the set of vertices w such that {u, w} is blue and {v, w} is red. Similarly, we define X RB u, v). Now since the solution is graphical X BR u, v) = X B u) \ X B u) X B v)) and X RB u, v) = X B v) \ X B u) X B v)), we get x BR u, v) x RB u, v) = X BR u, v) X RB u, v) = X B u) \ X B u) X B v)) X B v) \ X B u) X B v)) = X B u) X B u) X B v)) ) X B v) X B u) X B v)) ) = X B u) X B v) = x B u) x B v) proving 4). Set cu, v) = x RB u, v) + x BR u, v). Due to 4) and Furthermore, since we get Thus, x BR u, v) = cu, v) + x Bu) x B v) x RB u, v) = cu, v) x Bu) + x B v). 6) x R u) = n x B u) and x R v) = n x B v), x BR u, v)[x R u) + x B v)] + x RB u, v)[x B u) + x R v)] = cu, v) + [x Bu) x B v)] n [x B u) x B v)]) + cu, v) [x Bu) x B v)] n + [x B u) x B v)]) = cu, v)n [x B u) x B v)] cu, v)n. xbr u, v)[x R u) + x B v)] + x RB u, v)[x B u) + x R v)] ) Note that for any -coloring of the edges of K codegrb u, v) + codeg BR u, v) ) = w N n deg R w) deg B w), and since deg R w) deg B w) ) degr w) + deg B w) m ) = n, w [n] w [n] 5) cu, v). 7)

9 we obtain cu, v) = xrb u, v) + x BR u, v) ) m n/4. 8) Thus, 7) and 8) imply that λ 3 m, n) m n /4. Now we count the total number of alternating paths of length 4 with both endpoints in N. Each such path is of the form x 1 u x v x 3, where u, v [m] and x 1, x, x 3 [n]. Similarly as in the previous case we fix u, v [m] with u < v and count the number of paths going through u, v. Thus, the number of red-blue-red-blue paths is at most deg R u)codeg BR u, v) deg B v), we do not assume here that x 1, x and x 3 are different). Similarly the number of blue-red-blue-red paths is at most deg B u)codeg RB u, v) deg R v). Thus, the number of alternating path of length 4 is bounded from above by the solution of the following integer program: Maximize xr u)x BR u, v)x B v) + x B u)x RB u, v)x R v) ) subject to: The solution is graphical. As before we set cu, v) = x RB u, v) + x BR u, v) and by 5) and 6) we get x R u)x BR u, v)x B v) + x B u)x RB u, v)x R v) = = cu, v) + x Bu) x B v) [n x B u)]x B v) + cu, v) x Bu) + x B v) [n x B v)]x B u) = cu, v) xb u)[n x B u)] + x B v)[n x B v)] ) + 1 [x Bu) x B v)] [cu, v) n]. The last equality follows just from simple algebra operations. Since cu, v) n, the second term is at most 0. Furthermore, clearly x B u)[n x B u)] n /4 and x B v)[n x B v)] n /4. Thus, x R u)x BR u, v)x B v) + x B u)x RB u, v)x R v) cu, v)n /4.

10 PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE Consequently, by 8) xr u)x BR u, v)x B v) + x B u)x RB u, v)x R v) ) finishing the proof for path of length 4. n 4 cu, v) n 4 m n 4 = m n 3 16 Lemma 4.3 establishes the maximum number of alternating path of length 3 and 4. It is not difficult to see that in theory the approach taken in the proof of this lemma can be extended to count alternating paths of any length. For example, binding from above the number of paths of length 5 corresponds to the following integer program. Maximize fu, v, w) where subject to: 1 u<w<v m The solution is graphical. fu, w, v) = x BR u, w)x BR w, v)[x R u) + x B v)] + x RB u, w)x RB w, v)[x B u) + x R v)] + x RB u, w)x BR u, v)[x R w) + x B v)] + x BR u, w)x RB u, v)[x B w) + x R v)] + x BR u, v)x RB w, v)[x R u) + x B w)] + x RB u, v)x BR w, v)[x B u) + x R w)]. Unfortunately, the objective function is more complicated and its computations become more difficult and technical. One can also show that the problem of counting alternating paths in K = K m,n can be reduced to finding directed paths in a certain digraph. Indeed, let a -coloring of edges of K be given. Furthermore, assume that M N is the corresponding bipartition. We build a bipartite digraph D on M N as follows. For each red edge {u, v} with u N and v M we define a directed edge uv in D; otherwise, if {u, v} is blue, then we add to D directed edge vu. In other words, all red edges are oriented from N to M and blue ones from M to N. Clearly, each alternating path in K corresponds to a unique) directed path in D. Hence, we just reduced the problem of counting alternating paths in K to counting directed paths in bipartite tournament. Unfortunately, in general the problem of counting directed paths does not seem to be easy see, e.g., [10]). Finally let us mention that one can also study λ for any number r of colors, denoted by λ r,l m, n). Since λ r, m, n) = κ r, m, n), Theorem.1 implies that λ r, m, n) 1 1/r)m provided that n m log n. By assigning to the edges of K m,n colors from set [r] uniformly at random one can see like in the proof of Theorem 4.1) that λ r,k m, n) 1 + o1))m k n k 1 1 1/r) k 1 and λ k+1 m, n) 1 + o1))mn) k 1 1/r) k. The optimality of these bounds remains open.

5. Remarks on alternating connectivity Motivated by studying internally disjoint and alternating paths in a complete bipartite graph we introduce alternating connectivity. Let κ r,l G) be alternating connectivity of a graph G defined as maximum t such that there is an r-edge-coloring of G such that any pair of vertices is connected by t internally disjoint and alternating paths of length l. As in Sections and 3 we show that for complete graphs the following holds. Theorem 5.1. Let n 1. Then, for any number of colors r and for l 3 κ r, K n ) 1 1/r)n, 9) κ r,l K n ) n/l 1). 10) Proof. For 9) observe that like in the proof of Lemma. the total number of alternating paths of length in K n is at most due to the Cauchy-Schwarz inequality) deg i v) deg j v) = r 1 r deg i v)) deg i v) v V K n) 1 i<j r v V K n) i=1 i=1 = ) 1 r n 1) deg i v) r 1 r n 1) n, v V K n) and so κ r, K n ) r 1 r n 1) n / ) n = 1 1 ) n 1). r On the other hand, assigning colors from {1,..., r} uniformly at random to the edges of K n as in the proof of Lemma.3) yields κ r, K n ) 1 1/r)n on), provided that n 1. This finishes the proof of 9). For 10) we slightly modify proofs from Section 3. Clearly, the upper bound is trivial, since any collection of n/l 1) internally disjoint paths of length l must saturate all vertices. For the lower bound we will show that there exists a -coloring of the edges with the required property. First observe that one can easily generalize Lemma 3. for a truly) random graph Gn, 1/), where each edge is chosen independently with probability 1/. Lemma 5.. Let n 1 and 0 < α < 1/. Let Gn, 1/) be a random graph on set of vertices [n]. Then, w.h.p. for each A ) [n] αn and B [n] αn) with A B = there exists a matching between A and B of size αn on). Now we are ready to finish the proof of 10). Assign colors from {blue, red} uniformly at random to the edges of K n. Let u, v V K n ). By Chernoff s bound, N RR u, v) N RB u, v) N BR u, v) N BB u, v) n/4. Now if l is even, then we proceed like in the proof of Theorem 3.1. Let X B u) N BR u, v), X R v) N RR u, v) such that X B u) = X R v) = n/l 1)) on). In other words, all edges between u and X B u) are blue and all edges between v and X R v) are red. Now we choose disjoint subsets X 1,..., X l 3 N BR u, v) N RR u, v) \ i=1 11

1 PATRICK BENNETT, ANDRZEJ DUDEK, ELLIOT LAFORGE X B u) X R v)) such that X i = n/l 1)) + on). By Lemma 5. there is a red matching between X B u) and X 1, a blue matching between X 1 and X, etc., each of size n/l 1)) on). This yields n/l 1)) on) alternating and internally disjoint paths between u and v. To find the remaining n/l 1)) on) paths we choose X R u) N RBu, v), X B v) N BBu, v), and X 1,..., X l 3 N RBu, v) N BB u, v) \ X R u) X B v)). Finally, if l is odd, then we choose X B u) N BR u, v), X B v) N BB u, v), and X 1,..., X l 3 N BR u, v) N BB u, v) \ X B u) X B v)), and for the remaining paths X R u) N RBu, v), X R v) N RRu, v), and X 1,..., X l 3 N RBu, v) N RR u, v) \ X R u) X R v)). It might be of some interest to investigate κ r,l G) for an arbitrarily graph G. For example, studying alternating connectivity of random graphs could lead to better understanding of this concept. References [1] M. Albert, A. Frieze and B. Reed, Multicoloured Hamilton cycles, Electron. J. Combin. 1995), #R10. [] N. Alon and G. Gutin, Properly colored Hamiltonian cycles in edge-colored complete graphs, Random Structures Algorithms 11 1997), 179 186. [3] J. Bang-Jensen and G. Gutin, Alternating cycles and paths in edge-coloured multigraphs: a survey, Discrete Math. 165/166 1997), 39 60. [4] B. Bollobás and P. Erdős, Alternating Hamiltonian cycles, Israel J. Math. 3 1976), 16 131 [5] C. Chen and D. Daykin, Graphs with Hamiltonian cycles having adjacent lines different colors, J. Combin. Theory Ser. B 1 1976), 135 139. [6] A. Dudek and M. Ferrara, Extensions of results on rainbow Hamilton cycles in uniform hypergraphs, Graphs Combin. 31 015), no. 3, 577 583. [7] A. Dudek, A. Frieze and A. Ruciński, Rainbow Hamilton cycles in uniform hypergraphs, Electron. J. Combin. 19 01), no. 1, #46. [8] S. Janson, T. Luczak, and A. Ruciński, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 000. [9] V. Pless, Introduction to the theory of error-correcting codes, third ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1998. [10] P. Seymour and B.D. Sullivan, Counting paths in digraphs, European J. Combin. 31 010) 961 975. [11] J. Shearer, A property of the complete colored graph, Discrete Math. 5 1979), 175 178. Department of Mathematics, Western Michigan University, Kalamazoo, MI E-mail address: {patrick.bennett, andrzej.dudek, elliot.m.laforge}@wmich.edu