An upper bound on the distinguishing index of graphs with minimum degree at least two
aa r X i v : . [ m a t h . C O ] F e b An upper bound on the distinguishing index of graphs withminimum degree at least two
Saeid Alikhani ∗ Samaneh SoltaniJuly 21, 2018
Department of Mathematics, Yazd University, 89195-741, Yazd, Iran [email protected], [email protected]
Abstract
The distinguishing index of a simple graph G , denoted by D ′ ( G ), is the leastnumber of labels in an edge labeling of G not preserved by any non-trivial auto-morphism. It was conjectured by Pil´sniak (2015) that for any 2-connected graph D ′ ( G ) ≤ ⌈ p ∆( G ) ⌉ + 1. We prove a more general result for the distinguishing in-dex of graphs with minimum degree at least two from which the conjecture follows.Also we present graphs G for which D ′ ( G ) ≤ ⌈√ ∆ ⌉ . Keywords: distinguishing index; edge colourings; bound
AMS Subj. Class. : 05C25, 05C15
Let G = ( V, E ) be a simple connected graph. We use the standard graph notation ([4]).In particular, Aut( G ) denotes the automorphism group of G . For simple connectedgraph G , and v ∈ V , the neighborhood of a vertex v is the set N G ( v ) = { u ∈ V ( G ) : uv ∈ E ( G ) } . The degree of a vertex v in a graph G , denoted by deg G ( v ), is the numberof edges of G incident with v . In particular, deg G ( v ) is the number of neighbours of v in G . We denote by δ ( G ) and ∆( G ) the minimum and maximum degrees of the verticesof G . A graph G is k -regular if deg G ( v ) = k for all v ∈ V . The diameter of a graph G is the greatest distance between two vertices of G , and denoted by diam( G ).The distinguishing index D ′ ( G ) of a graph G is the least number d such that G hasan edge labeling with d labels that is preserved only by the identity automorphism of G . The distinguishing edge labeling was first defined by Kalinowski and Pi´sniak [6] forgraphs (was inspired by the well-known distinguishing number D ( G ) which was definedfor general vertex labelings by Albertson and Collins [1]). The distinguishing index ofsome examples of graphs was exhibited in [6]. For instance, D ′ ( P n ) = 2 for every n ≥ ∗ Corresponding author D ′ ( C n ) = 3 for n = 3 , , D ′ ( C n ) = 2 for n ≥
6. Also, for complete graphs K n ,we have D ′ ( K n ) = 3 for n = 3 , , D ′ ( K n ) = 2 for n ≥
6. They showed that if G isa connected graph of order n ≥ D ′ ( G ) ≤ ∆, unless G is C , C or C . It follows for connected graphs that D ′ ( G ) ≥ ∆( G ) if and only if D ′ ( G ) = ∆( G ) + 1 and G is a cycle of length at most five. The equality D ′ ( G ) = ∆( G )holds for all paths, for cycles of length at least 6, for K , K , and for symmetric orbisymmetric trees. Also, Pil´sniak showed that D ′ ( G ) < ∆( G ) for all other connectedgraphs. Theorem 1.1 [7]
Let G be a connected graph that is neither a symmetric nor an asym-metric tree. If the maximum degree of G is at least 3, then D ′ ( G ) ≤ ∆( G ) − unless G is K or K , . Pil´sniak put forward the following conjecture.
Conjecture 1.2 [7] If G is a -connected graph, then D ′ ( G ) ≤ ⌈ p ∆( G ) ⌉ . In this paper, we prove the following theorem which proves the conjecture.
Theorem 1.3
Let G be a connected graph of maximum degree ∆ . If the minimumdegree δ ≥ , then D ′ ( G ) ≤ ⌈√ ∆ ⌉ + 1 . For our purposes, we consider graphs with specific construction that are from dutch-windmill graphs. Because of this, in Section 2, we compute the distinguishing index ofthe dutch windmill graphs. In Section 3, we use the results to prove the main result.In the last section we present graphs G for which D ′ ( G ) ≤ ⌈√ ∆ ⌉ . To obtain the upper bound for the distinguishing index of connected graphs with mini-mum degree at least two, we characterize such graphs with minimum number of edges.For this characterization we need the concept of dutch windmill graphs. The dutchwindmill graph D kn is the graph obtained by taking n , ( n ≥
2) copies of the cycle graph C k , ( k ≥
3) with a vertex in common (see Figure 1). If k = 3, then we call D n , afriendship graph. In the following theorem we compute the distinguishing number ofdutch windmill graphs. Theorem 2.1
For every n ≥ and k ≥ , D ( D kn ) = min { r : r k − − r ⌈ k − ⌉ ≥ n } . Proof.
We consider two cases:
Case 1) If k is odd. There is a natural number m such that k = 2 m + 1. Wecan consider a blade of D kn as Figure 2. Let ( x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m , x ′ ( i ) m ) be the label ofvertices ( v , v ′ , . . . , v m , v ′ m ) of the i th blade where 1 ≤ i ≤ n . Suppose that L = { ( x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m , x ′ ( i ) m ) | ≤ i ≤ n, x ( i ) j , x ′ ( i ) j ∈ N , ≤ j ≤ m } is a labeling of thevertices of D kn except its central vertex. In an r -distinguishing labeling we must have:2igure 1: Examples of dutch windmill graphs.Figure 2: The considered polygon (or a cycle of size k ) in the proof of Theorem 2.1 .(i) There exists j ∈ { , . . . , m } such that x ( i ) j = x ′ ( i ) j for all i ∈ { , . . . , n } .(ii) For i = i we must have ( x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m , x ′ ( i ) m ) = ( x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m , x ′ ( i ) m )and ( x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m , x ′ ( i ) m ) = ( x ′ ( i )1 , x ( i )1 , . . . , x ′ ( i ) m , x ( i ) m ).There are r m − r m m )-arrays of labels using r labels satisfying (i) and (ii),and so D ( D kn ) = min { r : r m − r m ≥ n } . Case 2) If k is even. There is a natural number m such that k = 2 m . We canconsider a blade of D kn as Figure 2. Let ( x ( i )0 x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m − , x ′ ( i ) m − ) be the labelof vertices ( v , v , v ′ , . . . , v m − , v ′ m − ) of i th blade where 1 ≤ i ≤ n . Suppose that L = { ( x ( i )0 , x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m − , x ′ ( i ) m − ) | ≤ i ≤ n, x ( i )0 , x ( i ) j , x ′ ( i ) j ∈ N , ≤ j ≤ m − } is alabeling of the vertices of D kn except its central vertex. In an r -distinguishing labelingwe must have:(i) There exists j ∈ { , . . . , m − } such that x ( i ) j = x ′ ( i ) j for all i ∈ { , . . . , n } .3ii) For i = i we must have( x ( i )0 , x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m − , x ′ ( i ) m − ) = ( x ( i )0 , x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m − , x ′ ( i ) m − ) , ( x ( i )0 , x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m − , x ′ ( i ) m − ) = ( x ( i )0 , x ′ ( i )1 , x ( i )1 , . . . , x ′ ( i ) m − , x ( i ) m − ) . There are r m − − r m m − r labels satisfying (i)and (ii) ( r choices for x and r m − − r m − x ( i )1 , x ′ ( i )1 , . . . , x ( i ) m − , x ′ ( i ) m − ).Therefore D ( D kn ) = min { r : r m − − r m ≥ n } . (cid:3) The following theorem gives the distinguishing index of D kn . Theorem 2.2
For any n ≥ and k ≥ , D ′ ( D kn ) = min { r : r k − r ⌈ k ⌉ ≥ n } . Proof.
Since the effect of every automorphism of D k +1 n on its non-central vertices isexactly the same as the effect of an automorphism of D kn on its edges and vice versa,so if we consider the non-central vertices of D k +1 n as the edges of D kn , then we have D ′ ( D kn ) = D ( D k +1 n ). Therefore the result follows from Theorem 2.1. (cid:3) In this section, we shall prove Conjecture 1.2. To do this, first we state some preliminar-ies. By the result obtained by Fisher and Isaak [3] and independently by Imrich, Jerebicand Klavˇzar [5] the distinguishing index of complete bipartite graphs is as follows:
Theorem 3.1 [3, 5]
Let p, q, d be integers such that d ≥ and ( d − p < q ≤ d p .Then D ′ ( K p,q ) = (cid:26) d if q ≤ d p − ⌈ log d p ⌉ − ,d + 1 if q ≥ d p − ⌈ log d p ⌉ + 1 . If q = d p − ⌈ log d p ⌉ then the distinguishing index D ′ ( K p,q ) is either d or d + 1 andcan be computed recursively in O (log( q )) time. Corollary 3.2 [7] If p ≤ q , then D ′ ( K p,q ) ≤ ⌈ p √ q ⌉ + 1 . Also we need the following result:
Theorem 3.3 [7] If G is a graph of order n ≥ such that G has a Hamiltonian path,then D ′ ( G ) ≤ . Now, we state and prove the main theorem of this paper.
Theorem 3.4
Let G be a connected graph with maximum degree ∆ . If δ ≥ then D ′ ( G ) ≤ ⌈√ ∆ ⌉ . roof. If ∆ ≤
5, then the result follows from Theorem 1.1. So, we suppose that ∆ ≥ v be a vertex of G with the maximum degree ∆. By Theorem 2.2, we can labelthe edges of the dutch windmill graph attached to G at vertex v (a subgraph H isattached to graph G , if it has only one vertex in common with graph G ) for which v is the central point of the dutch windmill graph, with at most ⌈√ ∆ ⌉ labels from labelset { , , . . . , ⌈√ ∆ ⌉} , distinguishingly. If there exists triangle attached to G at v , thenwe label the two its incident edges to v with 0 and 1, and another edges of the trianglewith label 2.Let N (1) ( v ) = { v , . . . , v | N (1) ( v ) | } be the vertices of G at distance one from v , exceptthe vertices of dutch windmill or triangle attached to graph at v . We continue thelabeling by the following steps: Step 1)
Since | N (1) ( v ) | ≤ ∆, so for 0 ≤ i ≤ ⌈√ ∆ ⌉ and 1 ≤ j ≤ ⌈√ ∆ ⌉ −
1, welabel the edges vv i ⌈√ ∆ ⌉ + j with label i , and we do not use the label 0 any more. Withrespect to the number of incident edges to v with label 0, we conclude that the vertex v is fixed under each automorphism of G preserving the labeling. Also, since the dutchwindmill or the triangle graph attached to G at v has been labeled distinguishingly, sothe vertices of attached graph are fixed under each automorphism of G preserving thelabeling. Hence, every automorphism of G preserving the labeling must map the set ofvertices of G at distance i from v to itself setwise, for any 1 ≤ i ≤ diam( G ). We denotethe set of vertices of G at distance i from v , by N ( i ) ( v ) for 2 ≤ i ≤ diam( G ). If for any i ≥ N ( i ) ( v ) = ∅ , then G has a Hamiltonian path, and since ∆ ≥
6, so the order of G is at least 7, and hence D ′ ( G ) ≤ N ( i ) ( v ) = ∅ ,for some i ≥ N (1) ( v ) to two sets M (1)1 and M (1)2 as follows: M (1)1 = { x ∈ N (1) ( v ) : N ( x ) ⊆ N ( v ) } , M (1)2 = { x ∈ N (1) ( v ) : N ( x ) * N ( v ) } . The sets M (1)1 and M (1)2 are mapped to M (1)1 and M (1)2 , respectively, setwise, undereach automorphism of G preserving the labeling. For 0 ≤ i ≤ ⌈√ ∆ ⌉ , we set L i = { v i ⌈√ ∆ ⌉ + j : 1 ≤ j ≤ ⌈√ ∆ ⌉ − } . By this notation, we get that for 0 ≤ i ≤ ⌈√ ∆ ⌉ , theset L i is mapped to L i setwise, under each automorphism of G preserving the labeling.Let the sets M (1)1 i and M (1)2 i for 0 ≤ i ≤ ⌈√ ∆ ⌉ are as follows: M (1)1 i = M (1)1 ∩ L i , M (1)2 i = M (1)2 ∩ L i . It is clear that the sets M (1)1 i and M (1)2 i are mapped to M (1)1 i and M (1)2 i , respectively,setwise, under each automorphism of G preserving the labeling. Since for any 0 ≤ i ≤⌈√ ∆ ⌉ , we have | M (1)1 i | ≤ ⌈√ ∆ ⌉ −
1, so we can label all incident edges to each element of M (1)1 i with labels { , , . . . , ⌈√ ∆ ⌉} , such that for any two vertices of M (1)1 i , say x and y ,there exists a label k , 1 ≤ k ≤ ⌈√ ∆ ⌉ , such that the number of label k for the incidentedges to vertex x is different from the number of label k for the incident edges to vertex y . Hence, it can be deduce that each vertex of M (1)1 i is fixed under each automorphismof G preserving the labeling, where 0 ≤ i ≤ ⌈√ ∆ ⌉ . Thus every vertices of M (1)1 is fixed5nder each automorphism of G preserving the labeling. In sequel, we want to label theedges incident to vertices of M (1)2 such that M (1)2 is fixed under each automorphismof G preserving the labeling, pointwise. For this purpose, we partition the vertices of M (1)2 i to the sets M (1)2 i j , (1 ≤ j ≤ ∆ −
1) as follows: M (1)2 i j = { x ∈ M (1)2 i : | N ( x ) ∩ N (2) ( v ) | = j } . Since the set N ( i ) ( v ), for any i , is mapped to itself, it can be concluded that M (1)2 i j is mapped to itself, setwise, under each automorphism of G preserving the labeling, forany i and j . Let M (1)2 i j = { x i j , . . . , x i jsj } . It is clear that | M (1)2 i j | ≤ | M (1)2 i | ≤ ⌈√ ∆ ⌉ − x i jk ∈ M (1)2 i j , and N ( x i jk ) ∩ N (2) ( v ) = { x ′ i jk , x ′ i jk , . . . , x ′ i jkj } . We assign to the j -ary ( x i jk x ′ i jk , . . . , x i jk x ′ i jkj ) of edges, a j -ary of labels such that for every x i jk and x i jk ′ , 1 ≤ k, k ′ ≤ s j , there exists a label l in their corresponding j -arys of labels forwhich the number of label l in the corresponding j -arys of x i jk and x i jk ′ is distinct.For constructing | M (1)2 i j | numbers of such j -arys we need, min { r : (cid:0) j + r − r − (cid:1) ≥ | M (1)2 i j |} distinct labels. Since for any 1 ≤ j ≤ ∆ −
1, we havemin (cid:26) r : (cid:18) j + r − r − (cid:19) ≥ | M (1)2 i j | (cid:27) ≤ min (cid:26) r : (cid:18) j + r − r − (cid:19) ≥ ⌈√ ∆ ⌉ − (cid:27) ≤ ⌈√ ∆ ⌉ , so we need at most ⌈√ ∆ ⌉ distinct labels from label set { , , . . . , ⌈√ ∆ ⌉} for constructingsuch j -arys. For instance, let j = 1, and M (1)2 i = { x i , . . . , x i s } . By our method, welabel the edge x i x ′ i k with label k for 1 ≤ k ≤ s where s ≤ ⌈√ ∆ ⌉ . Hence, thevertices of M (1)2 i j , for any 1 ≤ j ≤ ∆ −
1, are fixed under each automorphism of G preserving the labeling. Therefore, the vertices of M (1)2 i for any 0 ≤ i ≤ ⌈√ ∆ ⌉ , and sothe vertices of M (1)2 are fixed under each automorphism of G preserving the labeling.Now, we can get that all vertices of N (1) ( v ) are fixed. If there exist unlabeled edges of G with the two endpoints in N (1) ( v ), then we assign them an arbitrary label, say 1. Step 2)
Now we consider N (2) ( v ). We partition this set such that the verticesof N (2) ( v ) with the same neighbours in M (1)2 , lie in a set. In other words, we canwrite N (2) ( v ) = S i A i , such that A i contains that elements of N (2) ( v ) having the sameneighbours in M (1)2 , for any i . Since all vertices in M (1)2 are fixed, so the set A i ismapped to A i setwise, under each automorphism of G preserving the labeling. Let A i = { w i , . . . , w it i } , and we have N ( w i ) ∩ M (1)2 = · · · = N ( w it i ) ∩ M (1)2 = { v i , . . . , v ip i } . We consider two following cases:Case 1) If for every w ij and w ij ′ in A i , where 1 ≤ j, j ′ ≤ t i , there exists a k ,1 ≤ k ≤ p i , for which the label of edge w ij v ik is different from label of edge w ij ′ v ik ,then all vertices of G in A i are fixed under each automorphism of G preserving thelabeling. 6ase 2) If there exist w ij and w ij ′ in A i , where 1 ≤ j, j ′ ≤ t i , such that for every k , 1 ≤ k ≤ p i , the label of edge w ij v ik and w ij ′ v ik are the same, then we can makea labeling such that the vertices in A i have the same property as Case 1, and so arefixed under each automorphism of G preserving the labeling, by using at least one ofthe following actions: • By commutating the coordinates of j -ary of labels assigned to the incident edgesto v ik with an end point in N (2) ( v ). • By using a new j -ary of labels, with labels { , , . . . , ⌈√ ∆ ⌉} , for incident edges to v ik with an end point in N (2) ( v ), such that (by notations in Step 1) for every x i ′ jk ′ and x i ′ jk ′′ , 1 ≤ k ′ , k ′′ ≤ s j , there exists a label l in their corresponding j -arys oflabels with different number of label l in their coordinates, where 1 ≤ i ′ ≤ ⌈√ ∆ ⌉ . • By labeling the unlabeled edges of G with the two end points in N (2) ( v ) whichare incident to the vertices in A i . • By labeling the unlabeled edges of G which are incident to the vertices in A i , andanother their endpoint is N (3) ( v ). • By labeling the unlabeled edges of G with the two end points in N (3) ( v ) for whichthe end points in N (3) ( v ) are adjacent to some of vertices in A i .Using at least one of above actions, it can be seen that every two vertices w ij and w ij ′ in A i have the property as Case 1. Thus we conclude that all vertices in A i , for any i , and so all vertices in N (2) ( v ), are fixed under each automorphism of G preserving thelabeling. If there exist unlabeled edges of G with the two endpoints in N (2) ( v ), thenwe assign them an arbitrary label, say 1.By continuing this method, in the next step we partition N (3) ( v ) exactly by thesame method as partition of N (2) ( v ) to the sets A i ’s in Step 2, and so we can make alabeling such that N ( i ) ( v ) is fixed pointwise, under each automorphism of G preservingthe labeling, for any 3 ≤ i ≤ diam( G ). (cid:3) For a 2-connected planar graph G , the distinguishing index may attain 1+ ⌈ p ∆( G ) ⌉ .For example, consider the complete bipartite graph K ,q with q = r , where r is apositive integer r . By Theorem 3.1, D ′ ( K ,q ) = r + 1. D ′ ( G ) ≤ ⌈√ ∆ ⌉ In this section, we present graphs G with specific construction such that D ′ ( G ) ≤ ⌈√ ∆ ⌉ .To do this we state the following definition. Definition 4.1
Let G be a connected graph with δ ( G ) ≥ . The graph G is called a δ -minimally graph , if the minimum degree of each spanning subgraph of G is less than δ ( G ) . v ).It can be concluded from Definition 4.1 that if e is an edge of a connected δ -minimally graph with end points u and w , then without loss of generality we canassume that deg G u = δ and deg G w ≥ δ . In fact the distance between the two verticesof degree greater than δ is at least two.The simplest connected 2-minimally graphs are cycles C n and complete bipartitegraphs K ,n . Now, we explain more on the structure of a 2-minimally graph. Let tocall a path in the graph a simple path , if all its internal vertices have degree two. Let G be a connected 2-minimally graph. • If the degree of all vertices of G is two, then G is a cycle graph. • If there exist a vertex v of G with degree at least three. We consider two followingcases:Case 1) If v is the only vertex of G with degree greater than two, then G is agraph which is made by identifying the central points of some dutch windmillgraphs D p i n i where p i ≥
3, and hence ∆( G ) = 2 P i ∈ I n i where I is a set of indices.In this case we denote G by Wind( v ) (for instance, see Figure 3).Case 2) If G has other vertex w of degree greater than two, then there exists atleast a simple path between v and w of length greater than one. Since G is aconnected graph, so if there exists no such simple path, then there exists a vertexof degree at least three on each path between v and w . Hence we can obtain avertex u of G with degree greater than two such that there exists at least a simplepath between v and u of length greater than one (see Figure 4).Now we characterize graphs G with D ′ ( G ) ≤ ⌈√ ∆ ⌉ . Theorem 4.2
Let G be a connected -minimally graph with maximum degree ∆ . If G is not a cycle C , C , C or a complete bipartite graph K ,r for some integer r , then D ′ ( G ) ≤ ⌈√ ∆ ⌉ . Proof.
If ∆ = 2, then G is a cycle. It is known that the distinguishing index of cyclegraph of order at least 6 is two. Hence, we suppose that G is not a cycle, so ∆ ≥ v be a vertex of G of maximum degree ∆. Suppose that V ′ = { v , v , . . . , v k } areall vertices of G which are of degree at least three such that there exists at least asimple path between v and v i , for any 1 ≤ i ≤ k (it is possible that V ′ = ∅ ). Let there8igure 4: The state of vertices of degree greater than two in a connected 2-minimallygraph.exist n ij disjoint simple paths of length j between v and v i , for any 1 ≤ i ≤ k and2 ≤ j ≤ diam( G ) where n ij is a non-negative integer and P diam( G ) j =2 n ij >
0. We canlabel these n ij simple paths of length j with at most ⌈√ ∆ ⌉ labels, by using n ij numbersof j -arys such that the coordinates of each j -ary are in the set { , , . . . , ⌈√ ∆ ⌉} , forany 1 ≤ i ≤ k and 2 ≤ j ≤ diam( G ), and for every two paths of length j , say P and P , there exists a label l , 1 ≤ l ≤ ⌈√ ∆ ⌉ , such that the number of label l in j -arysrelated to P and P are distinct. Let P be a simple path between v and v i for some1 ≤ i ≤ k , such that the label of edge of P which is incident to v , is different fromthe label of edge of P which is incident to v i . We do not use of labeling of the simplepath P , for any other simple path (with the same length) between any two verticesof degree greater than two. Since G is not a complete bipartite graph K ,r for someinteger r , so we can label these paths distinguishingly with at most ⌈√ ∆ ⌉ labels. Now,we label the induced subgraph Wind( x ), for any vertex x of degree greater than two, ifthere exists, with at most ⌈√ ∆ ⌉ labels distinguishingly by Theorem 2.2, such that thedistinguishing labeling of Wind( v ) is nonisomorphic to the remaining distinguishinglabeling of Wind( x ), where x ∈ V ( G ) − { v } . Thus any automorphism of G preservingthis labeling should be fixed v, v , . . . , v k and all vertices of degree two on the simplepaths between v and v i for any 1 ≤ i ≤ k . Since for any 1 ≤ i, j ≤ k where i = j , thevertices v i and v j are fixed, so all the simple paths between v i and v j , if there exist, aremapped to each other under each automorphism of G preserving this labeling. Hencewe can label all edges of these simple paths with at most √ ∆ labels, by assigningdistinct ordered arys of labels of length of the simple paths between v i and v j suchthat all vertices of these paths are fixed under each automorphism of G preserving thislabeling.For any 1 ≤ i ≤ k , we consider v i , and suppose that v i , . . . , v ik i are all vertices of V ( G ) \{ v , . . . , v k } with degree at least three such that there exists at least a simple pathbetween v i and v ij for any 1 ≤ j ≤ k i . Now we do the same method as labeling of simplepaths between v and { v , . . . , v k } , for all simple paths between v i and { v i , . . . , v ik i } with at most ⌈√ ∆ ⌉ labels. Also, we do the same method as labeling of simple pathsbetween v i and v j , for all simple paths between v ip and v iq with at most ⌈√ ∆ ⌉ labels,9here 1 ≤ p, q ≤ k i . Note that we do not use labeling of P for any simple path with thesame length as P between v i and { v i , . . . , v ik i } . Thus the vertices { v i , v i , . . . , v ik i } andall vertices of the simple paths between them are fixed under each automorphism of G preserving this labeling. After the finite number of steps we can obtain a distinguishingedge labeling of G with at most ⌈√ ∆ ⌉ labels. (cid:3) We gave an upper bound for the distinguishing index of graphs G with minimum degreeat least two. This result proves a conjecture by Pil´sniak (2015). We also studied graphs G with D ′ ( G ) ≤ ⌈√ ∆ ⌉ . We think that the following conjecture is true, but until nowall attempts to prove this failed. So, we end this paper by proposing the followingconjecture. Conjecture 5.1
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