A note on the neighbour-distinguishing index of digraphs
aa r X i v : . [ c s . D M ] S e p A note on the neighbour-distinguishingindex of digraphs ´Eric Sopena ∗ Mariusz Wo´zniak † September 24, 2019
Abstract
In this note, we introduce and study a new version of neighbour-distinguishingarc-colourings of digraphs. An arc-colouring γ of a digraph D is proper if no twoarcs with the same head or with the same tail are assigned the same colour. Foreach vertex u of D , we denote by S − γ ( u ) and S + γ ( u ) the sets of colours that ap-pear on the incoming arcs and on the outgoing arcs of u , respectively. An arccolouring γ of D is neighbour-distinguishing if, for every two adjacent vertices u and v of D , the ordered pairs ( S − γ ( u ) , S + γ ( u )) and ( S − γ ( v ) , S + γ ( v )) are distinct. Theneighbour-distinguishing index of D is then the smallest number of colours neededfor a neighbour-distinguishing arc-colouring of D .We prove upper bounds on the neighbour-distinguishing index of various classesof digraphs. Keywords:
Digraph; Arc-colouring; Neighbour-distinguishing arc-colouring.
MSC 2010:
A proper edge-colouring of a graph G is vertex-distinguishing if, for every two ver-tices u and v of G , the sets of colours that appear on the edges incident with u and v are distinct. Vertex-distinguishing proper edge-colourings of graphs were indepen-dently introduced by Burris and Schelp [2], and by ˇCerny, Horˇn´ak and Sot´ak [3].Requiring only adjacent vertices to be distinguished led to the notion of neighbour-distinguishing edge-colourings, considered in [1, 4, 7].Vertex-distinguishing arc-colourings of digraphs have been recently introducedand studied by Li, Bai, He and Sun [5]. An arc-colouring of a digraph is proper if ∗ Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR5800, F-33400 Talence, France. † AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Krakow, Poland. o two arcs with the same head or with the same tail are assigned the same colour.Such an arc-colouring is vertex-distinguishing if, for every two vertices u and v of G , (i) the sets S − ( u ) and S − ( v ) of colours that appear on the incoming arcs of u and v , respectively, are distinct, and (ii) the sets S + ( u ) and S + ( v ) of colours thatappear on the outgoing arcs of u and v , respectively, are distinct.In this paper, we introduce and study a neighbour-distinguishing version of arc-colourings of digraphs, using a slightly different distinction criteria: two neighbours u and v are distinguished whenever S − ( u ) = S − ( v ) or S + ( u ) = S + ( v ).Definitions and notation are introduced in the next section. We prove a generalupper bound on the neighbour-distinguishing index of a digraph in Section 3, andstudy various classes of digraphs in Section 4. Concluding remarks are given inSection 5. All digraphs we consider are without loops and multiple arcs. For a digraph D , wedenote by V ( D ) and A ( D ) its sets of vertices and arcs, respectively. The underlyinggraph of D , denoted und( D ), is the simple undirected graph obtained from D byreplacing each arc uv (or each pair of arcs uv , vu ) by the edge uv .If uv is an arc of a digraph D , u is the tail and v is the head of uv . For every vertex u of D , we denote by N + D ( u ) and N − D ( u ) the sets of out-neighbours and in-neighbours of u , respectively. Moreover, we denote by d + D ( u ) = | N + D ( u ) | and d − D ( u ) = | N − D ( u ) | the outdegree and indegree of u , respectively, and by d D ( u ) = d + D ( u ) + d − D ( u ) the degree of u .For a digraph D , we denote by δ + ( D ), δ − ( D ), ∆ + ( D ) and ∆ − ( D ) the minimumoutdegree, minimum indegree, maximum outdegree and maximum indegree of D ,respectively. Moreover, we let∆ ∗ ( D ) = max { ∆ + ( D ) , ∆ − ( D ) } . A (proper) k -arc-colouring of a digraph D is a mapping γ from V ( D ) to a set of k colours (usually { , . . . , k } ) such that, for every vertex u , (i) any two arcs with head u are assigned distinct colours, and (ii) any two arcs with tail u are assigned distinctcolours. Note here that two consecutive arcs vu and uw , v and w not necessarilydistinct, may be assigned the same colour. The chromatic index χ ′ ( D ) of a digraph D is then the smallest number k for which D admits a k -arc-colouring.The following fact is well-known (see e.g. [5, 6, 8]). Proposition 1
For every digraph D , χ ′ ( D ) = ∆ ∗ ( D ) . For every vertex u of a digraph D , and every arc-colouring γ of D , we denote by S + γ ( u ) and S − γ ( u ) the sets of colours assigned by γ to the outgoing and incoming arcsof u , respectively. From the definition of an arc-colouring, we get d + D ( u ) = | S + γ ( u ) | and d − D ( u ) = | S − γ ( u ) | for every vertex u . e say that two vertices u and v of a digraph D are distinguished by an arc-colouring γ of D , if ( S + γ ( u ) , S − γ ( u )) = ( S + γ ( v ) , S − γ ( v )). Note that we consider hereordered pairs, so that ( A, B ) = ( B, A ) whenever A = B . Note also that if u and v are such that d + D ( u ) = d + D ( v ) or d − D ( u ) = d − D ( v ), which happens in particular if d D ( u ) = d D ( v ), then they are distinguished by every arc-colouring of D . We willwrite u ≁ γ v if u and v are distinguished by γ and u ∼ γ v otherwise.A k -arc-colouring γ of a digraph D is neighbour-distinguishing if u ≁ γ v forevery arc uv ∈ A ( D ). Such an arc-colouring will be called an nd-arc-colouring forshort. The neighbour-distinguishing index ndi( D ) of a digraph D is then the smallestnumber of colours required for an nd-arc-colouring of D .The following lower bound is easy to establish. Proposition 2
For every digraph D , ndi( D ) ≥ χ ′ ( D ) = ∆ ∗ ( D ) . Moreover, if thereare two vertices u and v in D with d + D ( u ) = d + D ( v ) = d − D ( u ) = d − D ( v ) = ∆ ∗ ( D ) , then ndi( D ) ≥ ∆ ∗ ( D ) + 1 . Proof.
The first statement follows from the definitions. For the second statement,observe that S + γ ( u ) = S + γ ( v ) = S − γ ( u ) = S − γ ( v ) = { , . . . , ∆ ∗ ( D ) } for any two suchvertices u and v and any ∆ ∗ ( D )-arc-colouring γ of D . (cid:3) If D is an oriented graph , that is, a digraph with no opposite arcs, then every properedge-colouring ϕ of und( D ) is an nd-arc-colouring of D since, for every arc uv in D , ϕ ( uv ) ∈ S + ϕ ( u ) and ϕ ( uv ) / ∈ S + ϕ ( v ), which implies u ≁ ϕ v . Hence, we get thefollowing upper bound for oriented graphs, thanks to classical Vizing’s bound. Proposition 3 If D is an oriented graph, then ndi( D ) ≤ χ ′ (und( D )) ≤ ∆(und( D )) + 1 ≤ ∗ ( D ) + 2 . However, a proper edge-colouring of und( D ) may produce an arc-colouring of D which is not neighbour-distinguishing when D contains opposite arcs. Consider forinstance the digraph D given by V ( D ) = { a, b, c, d } and A ( D ) = { ab, bc, cb, dc } . Wethen have und( D ) = P , the path of order 4, and thus χ ′ (und( D )) = 2. It is thennot difficult to check that for any 2-edge-colouring ϕ of und( D ), S + ϕ ( b ) = S + ϕ ( c ) and S − ϕ ( b ) = S − ϕ ( c ).We will prove that the upper bound given in Proposition 3 can be decreased to2∆ ∗ ( D ), even when D contains opposite arcs. Recall that a digraph D is k -regular if d + D ( v ) = d − D ( v ) = k for every vertex v of D . A k -factor in a digraph D is aspanning k -regular subdigraph of D . The following result is folklore. Theorem 4
Every k -regular digraph can be decomposed into k arc-disjoint -factors. e first determine the neighbour-distinguishing index of a 1-factor. Proposition 5 If D is a digraph with d + D ( u ) = d − D ( u ) = 1 for every vertex u of D ,then ndi( D ) = 2 . Proof.
Such a digraph D is a disjoint union of directed cycles and any such cycleneeds at least two colours to be neighbour-distinguished. An nd-arc-colouring of D using two colours can be obtained as follows. For a directed cycle of even length, usealternately colours 1 and 2. For a directed cycle of odd length, use the colour 2 onany two consecutive arcs, and then use alternately colours 1 and 2. The so-obtained2-arc-colouring is clearly neighbour-distinguishing, so that ndi( D ) = 2. (cid:3) We are now able to prove the following general upper bound on the neighbour-distinguishing index of a digraph.
Theorem 6
For every digraph D , ndi( D ) ≤ ∗ ( D ) . Proof.
Let D ′ be any ∆ ∗ ( D )-regular digraph containing D as a subdigraph. If D is not already regular, such a digraph can be obtained from D by adding new arcs,and maybe new vertices.By Theorem 4, the digraph D ′ can be decomposed into ∆ ∗ ( D ′ ) = ∆ ∗ ( D ) arc-disjoint 1-factors, say F , . . . , F ∆ ∗ ( D ) . By Proposition 5, we know that D ′ admits annd-arc-colouring γ ′ using 2∆ ∗ ( D ′ ) = 2∆ ∗ ( D ) colours. We claim that the restriction γ of γ ′ to A ( D ) is also neighbour-distinguishing.To see that, let uv be any arc of D , and let t and w be the two vertices such thatthe directed walk tuvw belongs to a 1-factor F i of D ′ for some i , 1 ≤ i ≤ ∆ ∗ ( D ).Note here that we may have t = w , or w = u and t = v . If γ ( uv ) = γ ′ ( vw ), then γ ( uv ) ∈ S + γ ( u ) and γ ( uv ) / ∈ S + γ ( v ). Similarly, if γ ′ ( tu ) = γ ( uv ), then γ ( uv ) ∈ S − γ ( v )and γ ( uv ) / ∈ S − γ ( u ). Since neither three consecutive arcs nor two opposite arcs in awalk of a 1-factor of D ′ are assigned the same colour by γ ′ , we get that u ≁ γ v forevery arc uv of D , as required.This completes the proof. (cid:3) We study in this section the neighbour-distinguishing index of several classes ofdigraphs, namely complete symmetric digraphs, bipartite digraphs and digraphswhose underlying graph is k -chromatic, k ≥ We denote by K ∗ n the complete symmetric digraph of order n . Observe first thatany proper edge-colouring ǫ of K n induces an arc-colouring γ of K ∗ n defined by ( uv ) = γ ( vu ) = ǫ ( uv ) for every edge uv of K n . Moreover, since S + γ ( u ) = S − γ ( u ) = S ǫ ( u ) for every vertex u , γ is neighbour-distinguishing whenever ǫ is neighbour-distinguishing. Using a result of Zhang, Liu and Wang (see Theorem 6 in [7]), weget that ndi( K ∗ n ) = ∆ ∗ ( K ∗ n ) + 1 = n if n is odd, and ndi( K ∗ n ) ≤ ∆ ∗ ( K ∗ n ) + 2 = n + 1if n is even.We prove that the bound in the even case can be decreased by one (we recallthe proof of the odd case to be complete). Theorem 7
For every integer n ≥ , ndi( K ∗ n ) = ∆ ∗ ( K ∗ n ) + 1 = n . Proof.
Note first that we necessarily have ndi( K ∗ n ) ≥ n for every n ≥ V ( K ∗ n ) = { v , . . . , v n − } . If n = 2, we obviously have ndi( K ∗ ) = | A ( K ∗ ) | = 2 and the result follows. We can thus assume n ≥
3. We consider twocases, depending on the parity of n .Suppose first that n is odd, and consider a partition of the set of edges of K n into n disjoint maximal matchings, say M , . . . , M n − , such that for each i ,0 ≤ i ≤ n −
1, the matching M i does not cover the vertex v i . We define an n -arc-colouring γ of K ∗ n (using the set of colours { , . . . , n − } ) as follows. For every i and j , 0 ≤ i < j ≤ n −
1, we set γ ( v i v j ) = γ ( v j v i ) = k if and only if the edge v i v j belongs to M k . Observe now that for every vertex v i , 0 ≤ i ≤ n −
1, the colour i isthe unique colour that does not belong to S + γ ( v i ) ∪ S − γ ( v i ), since v i is not coveredby the matching M i . This implies that γ is an nd-arc-colouring of K ∗ n , and thusndi( K ∗ n ) = n , as required.Suppose now that n is even. Let K ′ be the subgraph of K ∗ n induced by the setof vertices { v , . . . , v n − } and γ ′ be the ( n − K ′ defined as above.We define an n -arc-colouring γ of K ∗ n (using the set of colours { , . . . , n − } ) asfollows:1. for every i and j , 0 ≤ i < j ≤ n − j i + 1 (mod n − γ ( v i v j ) = γ ′ ( v i v j ),2. for every i , 0 ≤ i ≤ n −
2, we set γ ( v i v i +1 ) = n − γ ( v i +1 v i ) = γ ′ ( v i +1 v i )(subscripts are taken modulo n − i , 0 ≤ i ≤ n −
2, we set γ ( v n − v i ) = γ ′ ( v i − v i ) and γ ( v i v n − ) = γ ′ ( v i +1 v i ).Since the colour n belongs to S + γ ( v i ) ∩ S − γ ( v i ) for every i , 0 ≤ i ≤ n −
2, and does notbelong to S + γ ( v n − ) ∪ S − γ ( v n − ), the vertex v n − is distinguished from every othervertex in K ∗ n . Moreover, for every vertex v i , 0 ≤ i ≤ n − S + γ ( v i ) = S + γ ′ ( v i ) ∪ { n − } and S − γ ( v i ) = S − γ ′ ( v i ) ∪ { n − } , which implies that any two vertices v i and v j , 0 ≤ i < j ≤ n −
2, are distinguishedsince γ ′ is an nd-arc-colouring of K ′ . We thus get that γ is an nd-arc-colouring of K ∗ n , and thus ndi( K ∗ n ) ≤ n , as required.This completes the proof. (cid:3) .2 Bipartite digraphs A digraph D is bipartite if its underlying graph is bipartite. In that case, V ( D ) = X ∪ Y with X ∩ Y = ∅ and A ( D ) ⊆ X × Y ∪ Y × X . We then have the followingresult. Theorem 8 If D is a bipartite digraph, then ndi( D ) ≤ ∆ ∗ ( D ) + 2 . Proof.
Let V ( D ) = X ∪ Y be the bipartition of V ( D ) and γ be any (not necessarilyneighbour-distinguishing) optimal arc-colouring of D using ∆ ∗ ( D ) colours (such anarc-colouring exists by Proposition 1).If γ is an nd-arc-colouring we are done. Otherwise, let M ⊆ A ( D ) ∩ ( X × Y )be a maximal matching from X to Y . We define the arc-colouring γ as follows: γ ( uv ) = ∆ ∗ ( D ) + 1 if uv ∈ M , γ ( uv ) = γ ( uv ) otherwise.Note that if uv is an arc such that u or v is (or both are) covered by M , then u ≁ γ v since the colour ∆ ∗ ( D ) + 1 appears in exactly one of the sets S + γ ( u ) and S + γ ( v ), or in exactly one of the sets S − γ ( u ) and S − γ ( v ).If γ is an nd-arc-colouring we are done. Otherwise, let A ∼ be the set of arcs uv ∈ A ( D ) with u ∼ γ v and M ⊆ A ∼ ∩ ( Y × X ) be a maximal matching from Y to X of A ∼ . We define the arc-colouring γ as follows: γ ( uv ) = ∆ ∗ ( D ) + 2 if uv ∈ M , γ ( uv ) = γ ( uv ) otherwise.Again, note that if uv is an arc such that u or v is (or both are) covered by M ,then u ≁ γ v . Moreover, since M is a matching of A ∼ , pairs of vertices that weredistinguished by γ are still distinguished by γ .Hence, every arc uv such that u and v were not distinguished by γ are nowdistinguished by γ which is thus an nd-arc-colouring of D using ∆ ∗ ( D ) + 2 colours.This concludes the proof. (cid:3) The upper bound given in Theorem 8 can be decreased when the underlyinggraph of D is a tree. Theorem 9 If D is a digraph whose underlying graph is a tree, then ndi( D ) ≤ ∆ ∗ ( D ) + 1 . Proof.
The proof is by induction on the order n of D . The result clearly holds if n ≤
2. Let now D be a digraph of order n ≥
3, such that the underlying graphund( D ) of D is a tree, and P = v . . . v k , k ≤ n , be a path in und( D ) with maximallength. By the induction hypothesis, there exists an nd-arc-colouring γ of D − v k using at most ∆ ∗ ( D − v k ) + 1 colours. We will extend γ to an nd-arc-colouring of D using at most ∆ ∗ ( D ) + 1 colours.If ∆ ∗ ( D ) = ∆ ∗ ( D − v k ) + 1, we assign the new colour ∆ ∗ ( D ) + 1 to the at mosttwo arcs incident with v k so that the so-obtained arc-colouring is clearly neighbour-distinguishing. uppose now that ∆ ∗ ( D ) = ∆ ∗ ( D − v k ). If all neighbours of v k − are leaves,the underlying graph of D is a star. In that case, there is at most one arc linking v k − and v k , and colouring this arc with any admissible colour produces an nd-arc-colouring of D . If the underlying graph of D is not a star, then, by the maximalityof P , we get that v k − has exactly one neighbour which is not a leaf, namely v k − .This implies that the only conflict that might appear when colouring the arcs linking v k and v k − is between v k − and v k − (recall that two neighbours with distinctindegree or outdegree are necessarily distinguished).Since d + D ( v k − ) ≤ ∆ ∗ ( D ) and d − D ( v k − ) ≤ ∆ ∗ ( D ), there necessarily exist a colour a such that S + γ ( v k − ) = S + γ ( v k − ) ∪ { a } , and a colour b such that S − γ ( v k − ) = S − γ ( v k − ) ∪ { b } . Therefore, the at most two arcs incident with v k can be coloured,using a and/or b , in such a way that the so-obtained arc-colouring is neighbour-distinguishing.This completes the proof. (cid:3) k -chromatic Since the set of edges of every k -colourable graph can be partitionned in ⌈ log k ⌉ parts each inducing a bipartite graph (see e.g. Lemma 4.1 in [1]), Theorem 8 leadsto the following general upper bound: Corollary 10 If D is a digraph whose underlying graph has chromatic number k ≥ , then ndi( D ) ≤ ∆ ∗ ( D ) + 2 ⌈ log k ⌉ . Proof.
Starting from an optimal arc-colouring of D with ∆ ∗ ( D ) colours, it sufficesto use two new colours for each of the ⌈ log k ⌉ bipartite parts (obtained from anyoptimal vertex-colouring of the underlying graph of D ), as shown in the proof ofTheorem 8, in order to get an nd-arc-colouring of D . (cid:3) In this note, we have introduced and studied a new version of neighbour-distingui-shing arc-colourings of digraphs. Pursuing this line of research, we propose thefollowing questions.1. Is there any general upper bound on the neighbour-distinguishing index ofsymmetric digraphs?2. Is there any general upper bound on the neighbour-distinguishing index of notnecessarily symmetric complete digraphs?3. Is there any general upper bound on the neighbour-distinguishing index ofdirected acyclic graphs? . The general bound given in Corollary 10 is certainly not optimal. In particular,is it possible to improve this bound for digraphs whose underlying graph is3-colourable?We finally propose the following conjecture. Conjecture 11
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