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Monochromatic connecting colorings in strongly connected oriented graphs
Discrete Mathematics 340 (2017) 578–584
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Monochromatic connecting colorings in strongly connected oriented graphs Diego González-Moreno a, *, Mucuy-kak Guevara b , Juan José Montellano-Ballesteros c a b c
Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana - Cuajimalpa, Mexico Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico
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Article history: Received 11 August 2015 Received in revised form 11 November 2016 Accepted 13 November 2016
Keywords: Monochromatic Strong connectivity Coloring Oriented graph
a b s t r a c t An arc-coloring of a strongly connected digraph D is a strongly monochromatic-connecting coloring if for every pair u, v of vertices in D there exist an (u, v )-monochromatic path and a (v, u)-monochromatic path. Let smc(D) denote the maximum number of colors that can be used in a strongly monochromatic-connecting coloring of D. In this paper we prove that if D is a strongly connected oriented graph of size m, then smc(D) = m − Ω (D) + 1, where Ω (D) is the minimum size of a spanning strongly connected subdigraph of D. As a corollary of this result, we see that a strongly connected oriented graph D of order n is hamiltonian if and only if smc(D) = m − n + 1. © 2016 Elsevier B.V. All rights reserved.
1. Introduction An interesting generalization of the concept of connectivity in graphs, due to Chartrand, Johns, McKeon and Zhang [5], is rainbow connecting colorings. An edge-colored graph G is rainbow connected if there exists a path, with no two edges colored the same, between any two vertices of G (see for instance [7]), and this concept can be easily extended to digraphs (see [1,6]). The rainbow connection number of a graph G is the minimum number of colors that are needed in order to make G rainbow connected. The rainbow connection number, besides being an interesting combinatorial measure, has applications to the secure transfer of classified information between agencies and in the area of networking. This applications background is from Li and Sun [7]. Caro and Yuster [4], as a naturally related question, introduced the concept of monochromatic-connecting coloring of a graph. An edge-coloring of a graph G is a monochromatic-connecting coloring if there exists a monochromatic path between any two vertices of G. The above definition can be also naturally extended to digraphs. An arc-coloring of a digraph D is a strongly monochromatic-connecting coloring (SMC coloring, for short) if for every pair u, v of vertices in D there exist an (u, v )-monochromatic path and a (v, u)-monochromatic path. The strong monochromatic connection number of a strongly connected digraph D, denoted by smc(D), is defined as the maximum number of colors used in an SMC coloring of D. Observe that given a strongly connected digraph D and a strongly connected spanning subdigraph H of D, by coloring the arcs of H with one single color and the remaining arcs with distinct colors, we obtain an SMC coloring of D with m −|A(H)|+ 1 colors, and therefore smc(D) ≥ m − |A(H)| + 1. In this paper the following theorem is proved.
*
Corresponding author. E-mail addresses: [email protected] (D. González-Moreno), [email protected] (M. Guevara), [email protected] (J.J. Montellano-Ballesteros). http://dx.doi.org/10.1016/j.disc.2016.11.016 0012-365X/© 2016 Elsevier B.V. All rights reserved.
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Theorem 1. Let D be a strongly connected oriented graph of size m, and let Ω (D) be the minimum size of a strongly connected spanning subdigraph of D. Then smc(D) = m − Ω (D) + 1. As a corollary of Theorem 1, it follows that an oriented graph D of order n is hamiltonian if and only if smc(D) = m − n + 1. From here we can see also that computing Ω (D) is NP-hard. 2. Definitions and notation All the digraphs considered in this work are finite oriented graphs, that is, a digraph with no symmetric arcs or loops. A digraph is said to be connected if its underlying graph is connected. Given a digraph D = (V (D), A(D)), a vertex v is reachable from a vertex u if D contains an (u, v )-path. A digraph D is unilateral if, for every pair u, v of vertices of D, either u is reachable from v or v is reachable from u (or both). A digraph D is strongly connected if for every pair of vertices {u, v} ⊆ V (D), the vertex u is reachable from v and the vertex v is reachable from u. Given a strongly connected digraph D, let Ω (D) denote the minimum size of a strongly connected spanning subdigraph of D. The minimum in-degree (resp. out-degree) of a subdigraph H of D will be denoted as δ − (H) (resp. δ + (H)). Let D = (V (D), A(D)) be a strongly connected digraph. Given S ⊆ V (D), the subdigraph induced by S is the subdigraph of D of maximum size which has S as set of vertices, and will be denoted as D[S ]. Given F ⊆ A(D), the subdigraph induced by F is the subdigraph of D of minimum order which has F as set of arcs, and will be denoted as D[F ]. Given a positive integer p, let [p] = {1, 2, . . . , p}. As a p-coloring of D we will understand a surjective function Γ : A(D) → [p]. For each ‘‘color’’ i ∈ [p] the set of arcs Γ −1 (i) will be called the chromatic class (of color i), and if |Γ −1 (i)| = 1, the color i and the chromatic class Γ −1 (i) will be called trivial. For each Γ , kΓ will be the number of non-trivial colors of Γ and, if kΓ ≥ 1, we will assume that [kΓ ] = {1, . . . , kΓ } ⊆ [p] is the set of non-trivial colors. A subdigraph H of D will be called monochromatic if A(H) is contained in a chromatic class, and we will say that the color i appears in H if A(H) ∩ Γ −1 (i) ̸ = ∅. A p-coloring of a digraph D is a strongly monochromatic-connecting coloring (SMC coloring, for short) if for every pair u, v of vertices in D there exist an (u, v )-monochromatic path and a (v, u)-monochromatic path. The strong monochromatic connection number of a strongly connected digraph D, denoted by smc(D), is defined as the maximum integer p such that there is a p-coloring of D which is an SMC coloring of D. We will say that a p-coloring Γ of D is a good coloring if p = smc(D) and Γ is an SMC coloring of D. For general concepts we may refer the reader to [2,3]. 3. The results In this section we present the proof of Theorem 1. We start with a remark and a lemma that will be used in the proof of Theorem 1. Remark 2. Let D = (V (D), A(D)) be a strongly connected oriented graph and let Γ be a SMC coloring of D. 1. Since D has no symmetric arcs, for each x ∈ V (D) there is at least one arc colored with a non-trivial color which is incident to x. 2. If Γ is a good coloring of D, for every non-trivial color i, D[Γ −1 (i)] is connected. 3. Let F ⊆ A(D) such that D[F ] is strongly connected and let I = {Γ (e) : e ∈ F }. Let V = V (D[F ]) and F ′ = A(D[V ]) \ F . Given i ∈ I, consider the coloring Γ ′ such that: Γ ′ (e) = Γ (e) for every e ∈ A(D) such that Γ (e) ̸ ∈ I; Γ ′ (e) = i for every e ∈ A(D) \ F ′ such that Γ (e) ∈ I; and each e ∈ F ′ such that Γ (e) ∈ I receive a trivial color. Γ ′ is a SMC coloring. Lemma 3. Let D = (V (D), A(D)) be a strongly connected oriented graph of order n. Then there is a good coloring Γ of D such that each non-trivial chromatic class of Γ induces a strongly connected subdigraph of D. Moreover, if Γ have at least 2 non-trivial chromatic classes, then each non-trivial chromatic class induces a subdigraph of order q, with 3 ≤ q ≤ n − 2. Proof. Let D = (V (D), A(D)) be a strongly connected oriented graph of order n, and let tD be the maximum number of trivial colors that appear in a good coloring of D. Given a good coloring Γ of D, for each non-trivial color i ∈ [kΓ ] let rΓ (i) be the number ∑ of ordered pairs of vertices (x, y) of the subdigraph D[Γ −1 (i)] such that there is no (x, y)-path in D[Γ −1 (i)], and let rΓ = i∈[kΓ ] rΓ (i). Let Γ be a good coloring of D with tD trivial colors and such that, among all the good colorings of D with tD trivial colors, rΓ is minimum. ⋃ For each i ∈ [kΓ ], let Hi = D[Γ −1 (i)] and let D∗ = D[ i∈[kΓ ] Γ −1 (i)]. Claim 1. If uv ∈ A(D∗ ) and P is an (u, v )-trail in D∗ − {uv}, then at least 3 colors appear in the subdigraph P ∪ {uv}. Let uv ∈ A(D∗ ) and P be an (u, v )-trail in D∗ − {uv}. Since uv ∈ A(D∗ ) it follows there is i ∈ [kΓ ] such that uv ∈ A(Hi ). If P is an (u, v )-trail in Hi − {uv}, the coloring Γ ′ obtained from Γ by assigning a new color to the arc uv is an SMC coloring of
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Fig. 1. SMC colorings Γ and Γ ′ .
D having more colors than Γ , which is a contradiction since Γ is a good coloring. So, at least two non-trivial colors appear in P ∪ {uv}. Suppose there are exactly two colors (i and j). Consider the coloring Γ ′ obtained from Γ by recoloring each arc of Γ −1 (i) \ {uv} with the color j. Observe that Γ ′ is a good coloring of D having more trivial colors than Γ , yielding a contradiction. □ Claim 2. For every i, with i ∈ [kΓ ], Hi is unilateral. Let i ∈ [kΓ ] and suppose Hi is not unilateral. Since Hi is connected, let {x, y} ⊆ V (Hi ) be a pair of vertices such that there is no (x, y)-path nor (y, x)-path in Hi , and with minimal distance between them in the underlying graph of Hi . If the distance is 2, there is z ∈ V (Hi ) such that either {xz , yz } ⊆ A(Hi ) or {zx, zy} ⊆ A(Hi ). In any case, since Γ is an SMC coloring, there is a monochromatic (x, y)-path and a monochromatic (y, x)-path, and at least one of them has a non trivial color j ̸ = i which, in any case, yields a contradiction with Claim 1. If the distance is r ≥ 3, let (x, w1 , . . . , wr −1 , y) be a (x, y)-path in the underlying graph of Hi , and suppose xw1 ∈ A(Hi ). Thus there is no (w1 , y)-path in Hi and by the minimality of the distance between x and y, there is a (y, w1 )-path (y, z1 , . . . , zq , w1 ) in Hi . Thus, there is no (zq , x)-path in Hi but since Γ is an SMC coloring there is a monochromatic (zq , x)-path P which, since is not of color i, by Claim 1 it must be of a trivial color h and therefore P is the arc zq x. Consider the coloring Γ ′ obtained from Γ by recoloring zq w1 with color color i to the arc zq x (see ∑ h and assigning∑ ′ Fig. 1). It is not hard to see that Γ ′ is a good coloring with tD trivial colors and i∈[kΓ ′ ] rΓ (Hi ) < i∈[kΓ ] rΓ (Hi ), yielding a contradiction with the election of Γ . In an analogous way we reach a contradiction for the case when w1 x ∈ A(Hi ) and the claim follows. □ Claim 3. Let i ∈ [kΓ ]. If max{δ + (Hi ), δ − (Hi )} ≥ 1, then Hi is a strongly connected subdigraph of D. Let i ∈ [kΓ ] and suppose δ + (Hi ) ≥ 1. By Claim 2 Hi is unilateral, thus, if Hi is not strongly connected, then there are two vertices u, v ∈ V (Hi ) such that there is a (u, v )-path in Hi and there is no (v, u)-path in Hi . Let T = (u = w1 , . . . , wl = v ) be an (u, v )-path in Hi . Since δ + (Hi ) ≥ 1, it is possible to extend T to a maximal path (u = w1 , . . . , wl = v, wl+1 , . . . , wr ) in Hi such that, for some s ∈ [r − 2], wr ws ∈ A(Hi ). Since there is no (v, u)-path in Hi , s ≥ 2 and since Γ is an SMC coloring, there exists a monochromatic (wr , u)-path P of color j, with j ̸ = i. Thus P ∪ (u = w1 , . . . , ws ) is a (wr , ws )-trail with colors i and j, and since wr ws have color i, by Claim 1 it follows that P is not contained in D∗ which implies that j is a trivial color and P is the arc wr u. Consider the coloring Γ ′ obtained from Γ by recoloring wr u with color ∑ i and assigning ∑color j to the arc wr ws (see Fig. 2). It is not hard to see that Γ ′ is a good coloring with tD trivial colors and i∈[k ′ ] rΓ ′ (Hi ) < i∈[kΓ ] rΓ (Hi ), yielding a Γ contradiction with the election of Γ . In an analogous way, if δ − (Hi ) ≥ 1 we reach a contradiction and the claim follows. □ Claim 4. Let i ∈ [kΓ ]. If max{δ + (Hi ), δ − (Hi )} = 0 then Hi is a directed path in D. Let i ∈ [kΓ ] such that max{δ + (Hi ), δ − (Hi )} = 0, and let v ∈ V (Hi ) with d− Hi (v ) = 0. Let T = (v = w1 , w2 , . . . , wr ) be a maximal directed path in Hi with starting vertex v . We will see that Hi = T . Observe that if for some pair of integers {s, t }, with 2 ≤ s < t ≤ r, there is a (wt , ws )-path P = (wt = z1 , . . . , zℓ = ws ) in Hi , by considering the path (v = w1 , w2 , . . . , wt = z1 , . . . , zℓ−1 ), the arc zℓ−1 zℓ = zℓ−1 ws and (since Γ is an SMC coloring and d− Hi (v ) = 0) the monochromatic (zℓ−1 , v )-path P of color j, with j ̸ = i, in the same way as in Claim 3, we will reach a contradiction. Also, by Claim 1 it follows there is no pair {s, t }, with 2 ≤ s < t ≤ r, such that ws wt ∈ A(Hi ), and, again by Claim 1, there is no s ∈ [r − 1] such that there is a (ws , ws+1 )-path in Hi − {ws ws+1 }. If Hi is not the path T , either there is a maximum j ∈ [r − 1] and x ∈ V (Hi ) \ V (T ) such that wj x ∈ A(Hi ), or there is a minimum j ∈ [r ] \ {1} and x ∈ V (Hi ) \ V (T ) such that xwj ∈ A(Hi ). If there is a maximum j ∈ [r − 1] and x ∈ V (Hi ) \ V (T ) such that wj x ∈ A(Hi ), since Γ is an SMC coloring there is a monochromatic (wj+1 , x)-path P in D and a monochromatic (x, wj+1 )-path Q in D. Since there is no (wj+1 , ws )-path in Hi (with s ≤ j) and since j is maximum and by Claim 1, it follows that P is the arc wj+1 x colored with a trivial color. Thus, Q is a monochromatic (x, wj+1 )-path in D colored with a non-trivial color and therefore Q is contained in D∗ . By Claim 1 we see that wj x ∪ Q is not a wj wj+1 -path contained in D∗ − wj wj+1 , thus wj wj+1 is in Q and therefore Q is contained in Hi , and since there is no (wj , ws )-path in Hi , with s < j, Q contains a xwj -path Q ′ (see Fig. 3). Consider the coloring Γ ′ obtained from Γ by recoloring wj+1 x with color i and assigning the trivial color of the arc wj+1 x to the arc wj x. It is not hard to see that Γ ′ is a
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Fig. 2. SMC colorings Γ and Γ ′ .
Fig. 3. wj+1 x is colored with a trivial color.
good coloring with tD trivial colors and, since j is maximum, i∈[k ′ ] rΓ ′ (Hi ) < i∈[kΓ ] rΓ (Hi ), yielding a contradiction with Γ the election of Γ . In an analogous way, if there is a minimum j ∈ [r ] \ {1} and x ∈ V (Hi ) \ V (T ) such that xwj ∈ A(Hi ) we reach a contradiction and the claim follows. □ By Claims 3 and 4 we see that for every i ∈ [kΓ ], either Hi is a strongly connected subdigraph or Hi is a directed path.
∑
∑
Claim 5. For every i ∈ [kΓ ], Hi is not a directed path. Let i ∈ [kΓ ] and suppose that Hi is the directed path (x1 , x2 , . . . , xr ). Since there are no symmetric arcs in D and Γ is an SMC coloring, for each i ∈ [r − 1] there is a monochromatic (xi+1 , xi )-path of a non-trivial color j. If Hj is strongly connected, there is a (xi , xi+1 )-path of color j which is a contradiction with Claim 1. Thus, Hj is another directed path. Let xa xa+1 be an arc of Hi and let P1 = (xa+1 = y11 , y12 , . . . , y1r1 = xa ) be a (xa+1 , xa )-path in D∗ . Consider the arc y1r −2 y1r −1 ∈ 1 1 A(P1 ) and let P2 = (y1r −1 = y21 , y22 , . . . , y2r2 = y1r −2 ) be a (y1r −1 , y1r −2 )-path in D∗ . Consider the arc y2r −2 y2r −1 ∈ A(P2 ) and let 1 1 1 1 2 2 P3 = (y2r −1 = y31 , y32 , . . . , y3r3 = y2r −2 ) be a (y2r −1 , y2r −2 )-path in D∗ . Continuing this process, since [kΓ ] is finite, it follows 2 2 2 2 that there exists a monochromatic (yℓrℓ −1 , yℓrℓ −2 )-path Pℓ+1 which is the first that uses the same color as one of the previous paths. Without loss of generality, suppose that Pℓ+1 ⊂ Hi , and that Pℓ+1 = (yℓrℓ −1 = xb , xb+1 , . . . , xc = yℓrℓ −2 ). Therefore (xa , xa+1 = y11 , y12 , . . . , y1r −2 , y1r −1 = y12 , y22 , . . . , y2r −2 , y2r −1 = y13 , y32 , . . . , yℓrℓ −2 , yℓrℓ −1 = xb , . . . , xc = yℓrℓ −2 ) is a trail in D∗ 1 1 2 2 (see Fig. 4). Let us distinguish two cases. Case 1. b ≤ a. In this case (xa , xa+1 = y11 , y12 , . . . , y1r −2 , y1r −1 = y12 , y22 , . . . , y2r −2 , y2r −1 = y13 , y32 , . . . , , yℓrℓ −2 , yℓrℓ −1 = xb , . . . , xa ) is a 1
1
2
2
j
j
closed trail. Consider the coloring Γ ′ obtained from Γ by the following way: for each j ∈ [ℓ], except for the arc yr −1 yrj j (which maintains its original color) all the other arcs of the chromatic class that contains the path Pj receive color i. Observe j j that in Γ ′ , for each j ∈ [ℓ] there is a (yr −1 , yrj )-path of color i. From here it is not difficult to see that Γ ′ is a good coloring of j D with more trivial colors than Γ , yielding a contradiction (see Fig. 5). Case 2. b > a. Since Γ is an SMC coloring, there exists a monochromatic (xc , xa )-path Q = (xc = v1 , . . . , vrx = xa ) which, since Hi is a path, has a different color than i. Thus, in this case (xa , xa+1 = y11 , y12 , . . . , y1r −2 , y1r −1 = y12 , y22 , . . . , y2r −2 , y2r −1 = 1 1 2 2 y13 , y32 , . . . , , yℓrℓ −2 , yℓrℓ −1 = xb , . . . , xc = v1 . . . , vrx = xa ) is a closed trail. Consider the coloring Γ ′ obtained from Γ in the following way: j j (i) For each j ∈ [ℓ − 1], except for the arc yr −1 yrj (which maintains its original color), all the other arcs of the chromatic class j that contain the path Pj receive the color i.
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Fig. 4. Structure of paths.
Fig. 5. Case 1.
Fig. 6. Case 2.
(ii) All the arcs of the chromatic class which contains Q receive color i and, except for the arc yℓrℓ −1 yℓrℓ (which maintains its original color) and for the arc xc xb = yℓrℓ −2 yℓrℓ −1 ∈ A(Pℓ ) (which receives the color of the path Q in Γ ), all the other arcs of the chromatic class that contains the path Pℓ receive color i. j j Observe that in Γ ′ , for each j ∈ [ℓ] there is a (yr −1 , yrj )-path of color i, and there is an (xc , xb )-path of color i. From here, j as in the previous case, we see that Γ ′ is a good coloring of D with more trivial colors than Γ , yielding a contradiction (see Fig. 6). □ By Claims 3–5 it follows that each non-trivial chromatic class of Γ induces a strongly connected subdigraph of D. Since there are no symmetric arcs in D, any strongly connected subdigraph has order at least 3. Let us suppose there are two non-trivial colors i, j of Γ and that |V (Hi )| ≥ n − 1. It is not hard to see that there is an arc xy ∈ A(Hj ) such that {x, y} ⊆ V (Hi ). Thus, since Hi is strongly connected, there is a (x, y)-path in Hi contradicting Claim 1, and the lemma follows. ■ Theorem 1.1. Let D be a strongly connected oriented graph of size m, and let Ω (D) be the minimum size of a strongly connected spanning subdigraph of D. Then smc(D) = m − Ω (D) + 1.
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Proof. Let D = (V (D), A(D)) be a strongly connected oriented graph of order n and size m. Observe that if n = 3 then D is a directed triangle, and it is not hard to see that Ω (D) = 3 and smc(D) = m − Ω (D) + 1 = 1. Let n ≥ 4. By Lemma 3 there is a good coloring Γ of D such that each non-trivial chromatic class induces a strongly connected subdigraph of D. Let q be the number of trivial colors in Γ , [k] be the set of non-trivial colors of Γ , and for each i ∈ [k], let Hi = D[Γ −1 (i)]. Observe that smc(D) = q + k. Given x ∈ V (D) let cx be the set of non-trivial colors i ∈ [k] such that x ∈ V (Hi ), and let Dx be the subdigraph of D induced ⋃ by i∈cx A(Hi ). Claim 6. For each x ∈ V (D), Dx is a strongly connected spanning subdigraph of D. Let y ∈ V (D). Since D has no symmetric arcs, either xy ̸ ∈ A(D) or yx ̸ ∈ A(D). Without loss of generality, suppose xy ̸ ∈ A(D). Since Γ is an SMC coloring, there is a monochromatic (x, y)-path P in D of a non-trivial color i ∈ [k]. Thus, since x ∈ V (P), i ∈ cx . Hence Hi is contained in Dx (and therefore P is contained in Dx ), and since Hi is strongly connected, Hi contains a monochromatic (y, x)-path of color i, and so does Dx . Thus, in Dx every vertex in V (D) is reachable from x, and x is reachable from any vertex of V (D). From here, the claim follows. □ If there is only one non-trivial color in Γ , then smc(D) = m − |A(H1 )| + 1, and for every x ∈ V (D), H1 = Dx . By Claim 6, H1 is a strongly connected spanning subdigraph of D and therefore |A(H1 )| ≥ Ω (D) and since smc(D) = m − |A(H1 )| + 1 the result follows. Thus, let us suppose there are k ≥ 2 non-trivial colors in Γ . Observe that by Lemma 3, this implies that for each i ∈ [k], 3 ≤ |V (Hi )| ≤ n − 2 and therefore n ≥ 5. Given x ∈ V (D), let Fx = A(D) \ A(Dx ). Since Dx is a strongly connected spanning subdigraph of D, |A(Dx )| ≥ Ω (D) and therefore m = |A(D)| = |A(Dx )| + |Fx | ≥ Ω (D) + |Fx |. Observe that Fx is the union of the trivial chromatic classes together with the set of arcs ∑ q + i∈[k]\cx |A(Hi )|. Adding over all the vertices in D we obtain that nm ≥ nΩ (D) +
∑
|Fx | = nΩ (D) + nq +
x∈V (D)
∑( ∑ x∈V (D)
⋃
i∈[k]\cx A(Hi ),
) |A(Hi )| .
thus |Fx | =
(1)
i∈[k]\cx
( The number of times the subdigraph Hi appears in the sum
∑
x∈V (D)
∑
) i∈[k]\cx |A(Hi )| is exactly n − |V (Hi )| (the number
of vertices y such that y ̸ ∈ V (Hi )), and since Hi is strongly connected, |A(Hi )| ≥ |V (Hi )|. Thus
∑( ∑ x∈V (D)
) ∑ ( ) |A(Hi )| ≥ |V (Hi )| n − |V (Hi )| .
i∈[k]\cx
i∈[k]
Therefore, by (1) nm ≥ nΩ (D) + nq +
∑
( ) |V (Hi )| n − |V (Hi )| .
i∈[k]
Dividing by n and adding 1 − Ω (D) on both sides of the inequality we obtain m − Ω (D) + 1 ≥ q + 1 +
( ) ∑ |V (Hi )| n − |V (Hi )| n
i∈[k]
.
In order to prove that smc(D) = m − Ω (D) + 1, since smc(D) = q + k it just remains to show that 1+
( ) ∑ |V (Hi )| n − |V (Hi )| i∈[k]
n
≥ k.
Since k ≥ 2, by Lemma 3, (it follows ) that for each i ∈ [k], 3 ≤ |V (Hi )| ≤ n − 2 and therefore, since n ≥ 5, it is not hard to see that for each i ∈ [k],
|V (Hi )| n−|V (Hi )| n
≥
2(n−2) n
≥ 1 and the result follows. ■
Corollary 4. Let D be a strongly connected oriented graph of size m and order n. D is hamiltonian if and only if smc(D) = m − n + 1. Proof. If D is hamiltonian, Ω (D) = n and therefore, by Theorem 1, smc(D) = m − n + 1. On the other hand, if smc(D) = m − n + 1, by Theorem 1 we see that Ω (D) = n which implies there is a strongly connected cycle of order n in D and the result follows. ■
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Acknowledgment The authors thank the referees for their kind observations. The first author’s research was supported by CONACyT-México, project CB-222104. The second author’s research was supported by PAPIIT México, project IN115816. The third author’s research was supported by PAPIIT México, project IN104915. References [1] [2] [3] [4] [5] [6] [7]
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Monochromatic connecting colorings in strongly connected oriented graphs Diego González-Moreno a, *, Mucuy-kak Guevara b , Juan José Montellano-Ballesteros c a b c
Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana - Cuajimalpa, Mexico Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico
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Article history: Received 11 August 2015 Received in revised form 11 November 2016 Accepted 13 November 2016
Keywords: Monochromatic Strong connectivity Coloring Oriented graph
a b s t r a c t An arc-coloring of a strongly connected digraph D is a strongly monochromatic-connecting coloring if for every pair u, v of vertices in D there exist an (u, v )-monochromatic path and a (v, u)-monochromatic path. Let smc(D) denote the maximum number of colors that can be used in a strongly monochromatic-connecting coloring of D. In this paper we prove that if D is a strongly connected oriented graph of size m, then smc(D) = m − Ω (D) + 1, where Ω (D) is the minimum size of a spanning strongly connected subdigraph of D. As a corollary of this result, we see that a strongly connected oriented graph D of order n is hamiltonian if and only if smc(D) = m − n + 1. © 2016 Elsevier B.V. All rights reserved.
1. Introduction An interesting generalization of the concept of connectivity in graphs, due to Chartrand, Johns, McKeon and Zhang [5], is rainbow connecting colorings. An edge-colored graph G is rainbow connected if there exists a path, with no two edges colored the same, between any two vertices of G (see for instance [7]), and this concept can be easily extended to digraphs (see [1,6]). The rainbow connection number of a graph G is the minimum number of colors that are needed in order to make G rainbow connected. The rainbow connection number, besides being an interesting combinatorial measure, has applications to the secure transfer of classified information between agencies and in the area of networking. This applications background is from Li and Sun [7]. Caro and Yuster [4], as a naturally related question, introduced the concept of monochromatic-connecting coloring of a graph. An edge-coloring of a graph G is a monochromatic-connecting coloring if there exists a monochromatic path between any two vertices of G. The above definition can be also naturally extended to digraphs. An arc-coloring of a digraph D is a strongly monochromatic-connecting coloring (SMC coloring, for short) if for every pair u, v of vertices in D there exist an (u, v )-monochromatic path and a (v, u)-monochromatic path. The strong monochromatic connection number of a strongly connected digraph D, denoted by smc(D), is defined as the maximum number of colors used in an SMC coloring of D. Observe that given a strongly connected digraph D and a strongly connected spanning subdigraph H of D, by coloring the arcs of H with one single color and the remaining arcs with distinct colors, we obtain an SMC coloring of D with m −|A(H)|+ 1 colors, and therefore smc(D) ≥ m − |A(H)| + 1. In this paper the following theorem is proved.
*
Corresponding author. E-mail addresses: [email protected] (D. González-Moreno), [email protected] (M. Guevara), [email protected] (J.J. Montellano-Ballesteros). http://dx.doi.org/10.1016/j.disc.2016.11.016 0012-365X/© 2016 Elsevier B.V. All rights reserved.
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Theorem 1. Let D be a strongly connected oriented graph of size m, and let Ω (D) be the minimum size of a strongly connected spanning subdigraph of D. Then smc(D) = m − Ω (D) + 1. As a corollary of Theorem 1, it follows that an oriented graph D of order n is hamiltonian if and only if smc(D) = m − n + 1. From here we can see also that computing Ω (D) is NP-hard. 2. Definitions and notation All the digraphs considered in this work are finite oriented graphs, that is, a digraph with no symmetric arcs or loops. A digraph is said to be connected if its underlying graph is connected. Given a digraph D = (V (D), A(D)), a vertex v is reachable from a vertex u if D contains an (u, v )-path. A digraph D is unilateral if, for every pair u, v of vertices of D, either u is reachable from v or v is reachable from u (or both). A digraph D is strongly connected if for every pair of vertices {u, v} ⊆ V (D), the vertex u is reachable from v and the vertex v is reachable from u. Given a strongly connected digraph D, let Ω (D) denote the minimum size of a strongly connected spanning subdigraph of D. The minimum in-degree (resp. out-degree) of a subdigraph H of D will be denoted as δ − (H) (resp. δ + (H)). Let D = (V (D), A(D)) be a strongly connected digraph. Given S ⊆ V (D), the subdigraph induced by S is the subdigraph of D of maximum size which has S as set of vertices, and will be denoted as D[S ]. Given F ⊆ A(D), the subdigraph induced by F is the subdigraph of D of minimum order which has F as set of arcs, and will be denoted as D[F ]. Given a positive integer p, let [p] = {1, 2, . . . , p}. As a p-coloring of D we will understand a surjective function Γ : A(D) → [p]. For each ‘‘color’’ i ∈ [p] the set of arcs Γ −1 (i) will be called the chromatic class (of color i), and if |Γ −1 (i)| = 1, the color i and the chromatic class Γ −1 (i) will be called trivial. For each Γ , kΓ will be the number of non-trivial colors of Γ and, if kΓ ≥ 1, we will assume that [kΓ ] = {1, . . . , kΓ } ⊆ [p] is the set of non-trivial colors. A subdigraph H of D will be called monochromatic if A(H) is contained in a chromatic class, and we will say that the color i appears in H if A(H) ∩ Γ −1 (i) ̸ = ∅. A p-coloring of a digraph D is a strongly monochromatic-connecting coloring (SMC coloring, for short) if for every pair u, v of vertices in D there exist an (u, v )-monochromatic path and a (v, u)-monochromatic path. The strong monochromatic connection number of a strongly connected digraph D, denoted by smc(D), is defined as the maximum integer p such that there is a p-coloring of D which is an SMC coloring of D. We will say that a p-coloring Γ of D is a good coloring if p = smc(D) and Γ is an SMC coloring of D. For general concepts we may refer the reader to [2,3]. 3. The results In this section we present the proof of Theorem 1. We start with a remark and a lemma that will be used in the proof of Theorem 1. Remark 2. Let D = (V (D), A(D)) be a strongly connected oriented graph and let Γ be a SMC coloring of D. 1. Since D has no symmetric arcs, for each x ∈ V (D) there is at least one arc colored with a non-trivial color which is incident to x. 2. If Γ is a good coloring of D, for every non-trivial color i, D[Γ −1 (i)] is connected. 3. Let F ⊆ A(D) such that D[F ] is strongly connected and let I = {Γ (e) : e ∈ F }. Let V = V (D[F ]) and F ′ = A(D[V ]) \ F . Given i ∈ I, consider the coloring Γ ′ such that: Γ ′ (e) = Γ (e) for every e ∈ A(D) such that Γ (e) ̸ ∈ I; Γ ′ (e) = i for every e ∈ A(D) \ F ′ such that Γ (e) ∈ I; and each e ∈ F ′ such that Γ (e) ∈ I receive a trivial color. Γ ′ is a SMC coloring. Lemma 3. Let D = (V (D), A(D)) be a strongly connected oriented graph of order n. Then there is a good coloring Γ of D such that each non-trivial chromatic class of Γ induces a strongly connected subdigraph of D. Moreover, if Γ have at least 2 non-trivial chromatic classes, then each non-trivial chromatic class induces a subdigraph of order q, with 3 ≤ q ≤ n − 2. Proof. Let D = (V (D), A(D)) be a strongly connected oriented graph of order n, and let tD be the maximum number of trivial colors that appear in a good coloring of D. Given a good coloring Γ of D, for each non-trivial color i ∈ [kΓ ] let rΓ (i) be the number ∑ of ordered pairs of vertices (x, y) of the subdigraph D[Γ −1 (i)] such that there is no (x, y)-path in D[Γ −1 (i)], and let rΓ = i∈[kΓ ] rΓ (i). Let Γ be a good coloring of D with tD trivial colors and such that, among all the good colorings of D with tD trivial colors, rΓ is minimum. ⋃ For each i ∈ [kΓ ], let Hi = D[Γ −1 (i)] and let D∗ = D[ i∈[kΓ ] Γ −1 (i)]. Claim 1. If uv ∈ A(D∗ ) and P is an (u, v )-trail in D∗ − {uv}, then at least 3 colors appear in the subdigraph P ∪ {uv}. Let uv ∈ A(D∗ ) and P be an (u, v )-trail in D∗ − {uv}. Since uv ∈ A(D∗ ) it follows there is i ∈ [kΓ ] such that uv ∈ A(Hi ). If P is an (u, v )-trail in Hi − {uv}, the coloring Γ ′ obtained from Γ by assigning a new color to the arc uv is an SMC coloring of
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Fig. 1. SMC colorings Γ and Γ ′ .
D having more colors than Γ , which is a contradiction since Γ is a good coloring. So, at least two non-trivial colors appear in P ∪ {uv}. Suppose there are exactly two colors (i and j). Consider the coloring Γ ′ obtained from Γ by recoloring each arc of Γ −1 (i) \ {uv} with the color j. Observe that Γ ′ is a good coloring of D having more trivial colors than Γ , yielding a contradiction. □ Claim 2. For every i, with i ∈ [kΓ ], Hi is unilateral. Let i ∈ [kΓ ] and suppose Hi is not unilateral. Since Hi is connected, let {x, y} ⊆ V (Hi ) be a pair of vertices such that there is no (x, y)-path nor (y, x)-path in Hi , and with minimal distance between them in the underlying graph of Hi . If the distance is 2, there is z ∈ V (Hi ) such that either {xz , yz } ⊆ A(Hi ) or {zx, zy} ⊆ A(Hi ). In any case, since Γ is an SMC coloring, there is a monochromatic (x, y)-path and a monochromatic (y, x)-path, and at least one of them has a non trivial color j ̸ = i which, in any case, yields a contradiction with Claim 1. If the distance is r ≥ 3, let (x, w1 , . . . , wr −1 , y) be a (x, y)-path in the underlying graph of Hi , and suppose xw1 ∈ A(Hi ). Thus there is no (w1 , y)-path in Hi and by the minimality of the distance between x and y, there is a (y, w1 )-path (y, z1 , . . . , zq , w1 ) in Hi . Thus, there is no (zq , x)-path in Hi but since Γ is an SMC coloring there is a monochromatic (zq , x)-path P which, since is not of color i, by Claim 1 it must be of a trivial color h and therefore P is the arc zq x. Consider the coloring Γ ′ obtained from Γ by recoloring zq w1 with color color i to the arc zq x (see ∑ h and assigning∑ ′ Fig. 1). It is not hard to see that Γ ′ is a good coloring with tD trivial colors and i∈[kΓ ′ ] rΓ (Hi ) < i∈[kΓ ] rΓ (Hi ), yielding a contradiction with the election of Γ . In an analogous way we reach a contradiction for the case when w1 x ∈ A(Hi ) and the claim follows. □ Claim 3. Let i ∈ [kΓ ]. If max{δ + (Hi ), δ − (Hi )} ≥ 1, then Hi is a strongly connected subdigraph of D. Let i ∈ [kΓ ] and suppose δ + (Hi ) ≥ 1. By Claim 2 Hi is unilateral, thus, if Hi is not strongly connected, then there are two vertices u, v ∈ V (Hi ) such that there is a (u, v )-path in Hi and there is no (v, u)-path in Hi . Let T = (u = w1 , . . . , wl = v ) be an (u, v )-path in Hi . Since δ + (Hi ) ≥ 1, it is possible to extend T to a maximal path (u = w1 , . . . , wl = v, wl+1 , . . . , wr ) in Hi such that, for some s ∈ [r − 2], wr ws ∈ A(Hi ). Since there is no (v, u)-path in Hi , s ≥ 2 and since Γ is an SMC coloring, there exists a monochromatic (wr , u)-path P of color j, with j ̸ = i. Thus P ∪ (u = w1 , . . . , ws ) is a (wr , ws )-trail with colors i and j, and since wr ws have color i, by Claim 1 it follows that P is not contained in D∗ which implies that j is a trivial color and P is the arc wr u. Consider the coloring Γ ′ obtained from Γ by recoloring wr u with color ∑ i and assigning ∑color j to the arc wr ws (see Fig. 2). It is not hard to see that Γ ′ is a good coloring with tD trivial colors and i∈[k ′ ] rΓ ′ (Hi ) < i∈[kΓ ] rΓ (Hi ), yielding a Γ contradiction with the election of Γ . In an analogous way, if δ − (Hi ) ≥ 1 we reach a contradiction and the claim follows. □ Claim 4. Let i ∈ [kΓ ]. If max{δ + (Hi ), δ − (Hi )} = 0 then Hi is a directed path in D. Let i ∈ [kΓ ] such that max{δ + (Hi ), δ − (Hi )} = 0, and let v ∈ V (Hi ) with d− Hi (v ) = 0. Let T = (v = w1 , w2 , . . . , wr ) be a maximal directed path in Hi with starting vertex v . We will see that Hi = T . Observe that if for some pair of integers {s, t }, with 2 ≤ s < t ≤ r, there is a (wt , ws )-path P = (wt = z1 , . . . , zℓ = ws ) in Hi , by considering the path (v = w1 , w2 , . . . , wt = z1 , . . . , zℓ−1 ), the arc zℓ−1 zℓ = zℓ−1 ws and (since Γ is an SMC coloring and d− Hi (v ) = 0) the monochromatic (zℓ−1 , v )-path P of color j, with j ̸ = i, in the same way as in Claim 3, we will reach a contradiction. Also, by Claim 1 it follows there is no pair {s, t }, with 2 ≤ s < t ≤ r, such that ws wt ∈ A(Hi ), and, again by Claim 1, there is no s ∈ [r − 1] such that there is a (ws , ws+1 )-path in Hi − {ws ws+1 }. If Hi is not the path T , either there is a maximum j ∈ [r − 1] and x ∈ V (Hi ) \ V (T ) such that wj x ∈ A(Hi ), or there is a minimum j ∈ [r ] \ {1} and x ∈ V (Hi ) \ V (T ) such that xwj ∈ A(Hi ). If there is a maximum j ∈ [r − 1] and x ∈ V (Hi ) \ V (T ) such that wj x ∈ A(Hi ), since Γ is an SMC coloring there is a monochromatic (wj+1 , x)-path P in D and a monochromatic (x, wj+1 )-path Q in D. Since there is no (wj+1 , ws )-path in Hi (with s ≤ j) and since j is maximum and by Claim 1, it follows that P is the arc wj+1 x colored with a trivial color. Thus, Q is a monochromatic (x, wj+1 )-path in D colored with a non-trivial color and therefore Q is contained in D∗ . By Claim 1 we see that wj x ∪ Q is not a wj wj+1 -path contained in D∗ − wj wj+1 , thus wj wj+1 is in Q and therefore Q is contained in Hi , and since there is no (wj , ws )-path in Hi , with s < j, Q contains a xwj -path Q ′ (see Fig. 3). Consider the coloring Γ ′ obtained from Γ by recoloring wj+1 x with color i and assigning the trivial color of the arc wj+1 x to the arc wj x. It is not hard to see that Γ ′ is a
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Fig. 2. SMC colorings Γ and Γ ′ .
Fig. 3. wj+1 x is colored with a trivial color.
good coloring with tD trivial colors and, since j is maximum, i∈[k ′ ] rΓ ′ (Hi ) < i∈[kΓ ] rΓ (Hi ), yielding a contradiction with Γ the election of Γ . In an analogous way, if there is a minimum j ∈ [r ] \ {1} and x ∈ V (Hi ) \ V (T ) such that xwj ∈ A(Hi ) we reach a contradiction and the claim follows. □ By Claims 3 and 4 we see that for every i ∈ [kΓ ], either Hi is a strongly connected subdigraph or Hi is a directed path.
∑
∑
Claim 5. For every i ∈ [kΓ ], Hi is not a directed path. Let i ∈ [kΓ ] and suppose that Hi is the directed path (x1 , x2 , . . . , xr ). Since there are no symmetric arcs in D and Γ is an SMC coloring, for each i ∈ [r − 1] there is a monochromatic (xi+1 , xi )-path of a non-trivial color j. If Hj is strongly connected, there is a (xi , xi+1 )-path of color j which is a contradiction with Claim 1. Thus, Hj is another directed path. Let xa xa+1 be an arc of Hi and let P1 = (xa+1 = y11 , y12 , . . . , y1r1 = xa ) be a (xa+1 , xa )-path in D∗ . Consider the arc y1r −2 y1r −1 ∈ 1 1 A(P1 ) and let P2 = (y1r −1 = y21 , y22 , . . . , y2r2 = y1r −2 ) be a (y1r −1 , y1r −2 )-path in D∗ . Consider the arc y2r −2 y2r −1 ∈ A(P2 ) and let 1 1 1 1 2 2 P3 = (y2r −1 = y31 , y32 , . . . , y3r3 = y2r −2 ) be a (y2r −1 , y2r −2 )-path in D∗ . Continuing this process, since [kΓ ] is finite, it follows 2 2 2 2 that there exists a monochromatic (yℓrℓ −1 , yℓrℓ −2 )-path Pℓ+1 which is the first that uses the same color as one of the previous paths. Without loss of generality, suppose that Pℓ+1 ⊂ Hi , and that Pℓ+1 = (yℓrℓ −1 = xb , xb+1 , . . . , xc = yℓrℓ −2 ). Therefore (xa , xa+1 = y11 , y12 , . . . , y1r −2 , y1r −1 = y12 , y22 , . . . , y2r −2 , y2r −1 = y13 , y32 , . . . , yℓrℓ −2 , yℓrℓ −1 = xb , . . . , xc = yℓrℓ −2 ) is a trail in D∗ 1 1 2 2 (see Fig. 4). Let us distinguish two cases. Case 1. b ≤ a. In this case (xa , xa+1 = y11 , y12 , . . . , y1r −2 , y1r −1 = y12 , y22 , . . . , y2r −2 , y2r −1 = y13 , y32 , . . . , , yℓrℓ −2 , yℓrℓ −1 = xb , . . . , xa ) is a 1
1
2
2
j
j
closed trail. Consider the coloring Γ ′ obtained from Γ by the following way: for each j ∈ [ℓ], except for the arc yr −1 yrj j (which maintains its original color) all the other arcs of the chromatic class that contains the path Pj receive color i. Observe j j that in Γ ′ , for each j ∈ [ℓ] there is a (yr −1 , yrj )-path of color i. From here it is not difficult to see that Γ ′ is a good coloring of j D with more trivial colors than Γ , yielding a contradiction (see Fig. 5). Case 2. b > a. Since Γ is an SMC coloring, there exists a monochromatic (xc , xa )-path Q = (xc = v1 , . . . , vrx = xa ) which, since Hi is a path, has a different color than i. Thus, in this case (xa , xa+1 = y11 , y12 , . . . , y1r −2 , y1r −1 = y12 , y22 , . . . , y2r −2 , y2r −1 = 1 1 2 2 y13 , y32 , . . . , , yℓrℓ −2 , yℓrℓ −1 = xb , . . . , xc = v1 . . . , vrx = xa ) is a closed trail. Consider the coloring Γ ′ obtained from Γ in the following way: j j (i) For each j ∈ [ℓ − 1], except for the arc yr −1 yrj (which maintains its original color), all the other arcs of the chromatic class j that contain the path Pj receive the color i.
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Fig. 4. Structure of paths.
Fig. 5. Case 1.
Fig. 6. Case 2.
(ii) All the arcs of the chromatic class which contains Q receive color i and, except for the arc yℓrℓ −1 yℓrℓ (which maintains its original color) and for the arc xc xb = yℓrℓ −2 yℓrℓ −1 ∈ A(Pℓ ) (which receives the color of the path Q in Γ ), all the other arcs of the chromatic class that contains the path Pℓ receive color i. j j Observe that in Γ ′ , for each j ∈ [ℓ] there is a (yr −1 , yrj )-path of color i, and there is an (xc , xb )-path of color i. From here, j as in the previous case, we see that Γ ′ is a good coloring of D with more trivial colors than Γ , yielding a contradiction (see Fig. 6). □ By Claims 3–5 it follows that each non-trivial chromatic class of Γ induces a strongly connected subdigraph of D. Since there are no symmetric arcs in D, any strongly connected subdigraph has order at least 3. Let us suppose there are two non-trivial colors i, j of Γ and that |V (Hi )| ≥ n − 1. It is not hard to see that there is an arc xy ∈ A(Hj ) such that {x, y} ⊆ V (Hi ). Thus, since Hi is strongly connected, there is a (x, y)-path in Hi contradicting Claim 1, and the lemma follows. ■ Theorem 1.1. Let D be a strongly connected oriented graph of size m, and let Ω (D) be the minimum size of a strongly connected spanning subdigraph of D. Then smc(D) = m − Ω (D) + 1.
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Proof. Let D = (V (D), A(D)) be a strongly connected oriented graph of order n and size m. Observe that if n = 3 then D is a directed triangle, and it is not hard to see that Ω (D) = 3 and smc(D) = m − Ω (D) + 1 = 1. Let n ≥ 4. By Lemma 3 there is a good coloring Γ of D such that each non-trivial chromatic class induces a strongly connected subdigraph of D. Let q be the number of trivial colors in Γ , [k] be the set of non-trivial colors of Γ , and for each i ∈ [k], let Hi = D[Γ −1 (i)]. Observe that smc(D) = q + k. Given x ∈ V (D) let cx be the set of non-trivial colors i ∈ [k] such that x ∈ V (Hi ), and let Dx be the subdigraph of D induced ⋃ by i∈cx A(Hi ). Claim 6. For each x ∈ V (D), Dx is a strongly connected spanning subdigraph of D. Let y ∈ V (D). Since D has no symmetric arcs, either xy ̸ ∈ A(D) or yx ̸ ∈ A(D). Without loss of generality, suppose xy ̸ ∈ A(D). Since Γ is an SMC coloring, there is a monochromatic (x, y)-path P in D of a non-trivial color i ∈ [k]. Thus, since x ∈ V (P), i ∈ cx . Hence Hi is contained in Dx (and therefore P is contained in Dx ), and since Hi is strongly connected, Hi contains a monochromatic (y, x)-path of color i, and so does Dx . Thus, in Dx every vertex in V (D) is reachable from x, and x is reachable from any vertex of V (D). From here, the claim follows. □ If there is only one non-trivial color in Γ , then smc(D) = m − |A(H1 )| + 1, and for every x ∈ V (D), H1 = Dx . By Claim 6, H1 is a strongly connected spanning subdigraph of D and therefore |A(H1 )| ≥ Ω (D) and since smc(D) = m − |A(H1 )| + 1 the result follows. Thus, let us suppose there are k ≥ 2 non-trivial colors in Γ . Observe that by Lemma 3, this implies that for each i ∈ [k], 3 ≤ |V (Hi )| ≤ n − 2 and therefore n ≥ 5. Given x ∈ V (D), let Fx = A(D) \ A(Dx ). Since Dx is a strongly connected spanning subdigraph of D, |A(Dx )| ≥ Ω (D) and therefore m = |A(D)| = |A(Dx )| + |Fx | ≥ Ω (D) + |Fx |. Observe that Fx is the union of the trivial chromatic classes together with the set of arcs ∑ q + i∈[k]\cx |A(Hi )|. Adding over all the vertices in D we obtain that nm ≥ nΩ (D) +
∑
|Fx | = nΩ (D) + nq +
x∈V (D)
∑( ∑ x∈V (D)
⋃
i∈[k]\cx A(Hi ),
) |A(Hi )| .
thus |Fx | =
(1)
i∈[k]\cx
( The number of times the subdigraph Hi appears in the sum
∑
x∈V (D)
∑
) i∈[k]\cx |A(Hi )| is exactly n − |V (Hi )| (the number
of vertices y such that y ̸ ∈ V (Hi )), and since Hi is strongly connected, |A(Hi )| ≥ |V (Hi )|. Thus
∑( ∑ x∈V (D)
) ∑ ( ) |A(Hi )| ≥ |V (Hi )| n − |V (Hi )| .
i∈[k]\cx
i∈[k]
Therefore, by (1) nm ≥ nΩ (D) + nq +
∑
( ) |V (Hi )| n − |V (Hi )| .
i∈[k]
Dividing by n and adding 1 − Ω (D) on both sides of the inequality we obtain m − Ω (D) + 1 ≥ q + 1 +
( ) ∑ |V (Hi )| n − |V (Hi )| n
i∈[k]
.
In order to prove that smc(D) = m − Ω (D) + 1, since smc(D) = q + k it just remains to show that 1+
( ) ∑ |V (Hi )| n − |V (Hi )| i∈[k]
n
≥ k.
Since k ≥ 2, by Lemma 3, (it follows ) that for each i ∈ [k], 3 ≤ |V (Hi )| ≤ n − 2 and therefore, since n ≥ 5, it is not hard to see that for each i ∈ [k],
|V (Hi )| n−|V (Hi )| n
≥
2(n−2) n
≥ 1 and the result follows. ■
Corollary 4. Let D be a strongly connected oriented graph of size m and order n. D is hamiltonian if and only if smc(D) = m − n + 1. Proof. If D is hamiltonian, Ω (D) = n and therefore, by Theorem 1, smc(D) = m − n + 1. On the other hand, if smc(D) = m − n + 1, by Theorem 1 we see that Ω (D) = n which implies there is a strongly connected cycle of order n in D and the result follows. ■
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Acknowledgment The authors thank the referees for their kind observations. The first author’s research was supported by CONACyT-México, project CB-222104. The second author’s research was supported by PAPIIT México, project IN115816. The third author’s research was supported by PAPIIT México, project IN104915. References [1] [2] [3] [4] [5] [6] [7]
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