aa r X i v : . [ m a t h . C O ] J un Discrete Mathematics and Theoretical Computer Science
DMTCS vol. :1, 2018, Group twin edge coloring of graphs
Sylwia Cichacz ∗ Jakub Przybyło † AGH University of Science and Technology received 2017-9-22 , revised 2018-2-19,2018-6-5 , accepted 2018-6-5 . For a given graph G , the least integer k ≥ such that for every Abelian group G of order k there exists a proper edgelabeling f : E ( G ) → G so that P x ∈ N ( u ) f ( xu ) = P x ∈ N ( v ) f ( xv ) for each edge uv ∈ E ( G ) is called the grouptwin chromatic index of G and denoted by χ ′ g ( G ) . This graph invariant is related to a few well-known problems inthe field of neighbor distinguishing graph colorings. We conjecture that χ ′ g ( G ) ≤ ∆( G ) + 3 for all graphs withoutisolated edges, where ∆( G ) is the maximum degree of G , and provide an infinite family of connected graph (trees)for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the basecase for proving a general upper bound for all graphs G without isolated edges: χ ′ g ( G ) ≤ G ) + col( G )) − ,where col( G ) denotes the coloring number of G . This improves the best known upper bound known previously onlyfor the case of cyclic groups Z k . Keywords:
Abelian group, twin edge coloring
It is a well-known fact that in any simple graph G there are at least two vertices of the same degree. Thesituation changes if we consider an edge labeling f : E ( G ) → { , . . . , s } and calculate the weighteddegree of each vertex v as the sum of labels of all the edges incident with v . The labeling f is called irregular if the weighted degrees of all the vertices in G are distinct. The least value of s that allows someirregular labeling is called the irregularity strength of G and denoted by s ( G ) .The problem of finding s ( G ) was introduced by Chartrand et al. in [CJL +
88] and investigated bynumerous authors [AT98, CL08, FSJL89, FGKP02, Leh91]. A tight upper bound s ( G ) ≤ n − , where n is the order of G , was proved for all graphs containing no isolated edges and at most one isolated vertex,except for the graph K [AT90, Nie00]. This was improved for graphs with sufficiently large minimumdegree δ by Kalkowski, Karo´nski and Pfender [KKP11], who proved that s ( G ) ≤ ⌈ n/δ ⌉ , and for graphswith δ ≥ n / ln n by Majerski and Przybyło in [MP14], implying that s ( G ) ≤ (4 + o (1)) n/δ + 4 then.A labeling of the edges of a graph G is called vertex coloring if it results in weighted degrees thatproperly color the vertices (i.e., weighted degrees are required to be distinct only for adjacent vertices). If ∗ This work was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministryof Science and Higher Education. † Supported by the National Science Centre, Poland, grant no. 2014/13/B/ST1/01855.ISSN subm. to DMTCS c (cid:13)
Sylwia Cichacz, Jakub Przybyło we use the elements of { , , . . . , k } to label the edges, such a labeling is called a vertex coloring k -edgelabeling .The concept of coloring the vertices with the sums of edge labels was introduced for the first time byKaro´nski, Łuczak and Thomason [KŁT04]. The authors posed the following question. Given a graph G without isolated edges, what is the minimum k such that there exists a vertex coloring k -edge labeling?We will call this minimum value of k the sum chromatic number and denote it by χ Σ ( G ) . Karo´nski,Łuczak and Thomason conjectured that χ Σ ( G ) ≤ for every graph G with no isolated edges. The firstconstant bound was proved by Addario-Berry et al. in [ABDM +
07] ( χ Σ ( G ) ≤ ) and then improvedby Addario-Berry et al. in [ABDR08] ( χ Σ ( G ) ≤ ), Wan and Yu in [WY08] ( χ Σ ( G ) ≤ ) and finallyby Kalkowski, Karo´nski and Pfender in [KKP10] ( χ Σ ( G ) ≤ ). Recently Thomassen, Wu and Zhangconsidered the modulo version of this problem [TWZ16]. Specifically, they proved that a non-bipartite (6 k − -edge-connected graph of chromatic number at most k admits a weighting of the edges withlabels , such that the resulting weighted degrees reduced modulo k yield a proper vertex coloring ofthe vertices. A variation of the sum chromatic number with labels from any Abelian group is called the group sum chromatic number and was studied in [AC16]; more precisely it is the least integer s such thatfor any Abelian group G of order s , there exists a function f : E ( G ) → G which induce a proper coloringof the vertices by their corresponding sums of incident labels. Such problem was in fact first consideredin [KŁT04].Inspired by the graph colorings described above, Andrews et al. [AHJ +
14] turned towards proper edgelabelings (with distinct labels on adjacent edges) with the elements of a given Z k . By a twin edge coloring of a graph G (without isolated edges) they denoted a proper edge labeling f : E ( G ) → Z k for some k ≥ such that the induced vertex coloring w : V ( G ) → Z k defined by w ( v ) = P u ∈ N ( v ) f ( uv ) (mod k ) isproper. The least integer k admitting such an edge labeling is called the twin chromatic index of G anddenoted by χ ′ t ( G ) . Note that since f constitutes a proper edge coloring, then ∆( G ) ≤ χ ′ ( G ) ≤ χ ′ t ( G ) .Andrews et al. showed that if G is a connected graph of order at least and size m , then χ ′ t ( G ) ≤ m − .They also stated the following conjecture and verified it for some classes of graphs: Conjecture 1.1 ([AHJ + If G is a connected graph of order at least that is not a -cycle, then χ ′ t ( G ) ≤ ∆( G ) + 2 . This was a strengthening of a former conjecture of Flandrin et al. [FMP +
13] with the same thesis butconcerning a protoplast of χ ′ t where instead of calculating appropriate sums modulo k , we simply computethese in Z . See [BP17, DWZ14, Prz13, PW15, WCW14] for other results concerning this graph invariant.In [AJZ14], Andrews et al. estimated the twin chromatic index for some classes of graphs; in particularthey proved the following theorem for trees with small maximum degree. Theorem 1.2 ([AJZ14]) If T is a tree of order at least and ∆( T ) ≤ , then T has a twin edge (∆( T ) +2) -coloring. Moreover if T is a path of order n ≥ , then χ ′ t ( T ) = 3 . For an integer r ≥ , a tree T is called r - regular if each non-leaf of T has degree r . Theorem 1.3 ([Joh15]) If T is a regular tree of order at least , then χ ′ t ( T ) ≤ ∆( T ) + 2 . Moreover if ∆( T ) ≡ then χ ′ t ( T ) = ∆( T ) + 2 . As for a general upper bound, the best thus far result is due to Johnston [Joh15], who proved thefollowing.
Theorem 1.4 ([Joh15]) If G is a connected graph of order at least , then χ ′ t ( G ) ≤ G ) − .roup twin edge coloring of graphs G is an Abelian group of order k with the operation denoted by + and the identity element .For convenience we will write ma to denote a + a + . . . + a (where element a appears m times), − a todenote the inverse of a and we will use a − b instead of a + ( − b ) . Moreover, the notation P a ∈ S a willbe used as a short form for a + a + a + . . . , where a , a , a , . . . are all the elements of the set S . Wewill call a proper edge labeling f : E ( G ) → G a G - twin edge coloring if the resulting weighted degrees ,defined for every vertex v ∈ V ( G ) as the sum (in G ): w ( v ) = X u ∈ N ( v ) f ( uv ) , yield a proper vertex coloring of G , i.e. we have w ( u ) = w ( v ) for every edge uv ∈ E ( G ) . We willalso call w ( v ) the color of a vertex v or the sum at v , while such a labeling f will be referred to as neighbor sum distinguishing as well. Generalizing the concept of the twin chromatic index, the leastinteger k ≥ for which G has a G -twin edge coloring for every Abelian group G of order k is called the group twin chromatic index of G and is denoted by χ ′ g ( G ) . Obviously χ ′ t ( G ) ≤ χ ′ g ( G ) for any graph G (without isolated edges), and there are plenty of graphs for which χ ′ t ( G ) < χ ′ g ( G ) (cf. Theorem 1.3 andObservation 3.4). Note here also that the fact that χ ′ g ( G ) ≤ K for a given graph G and a constant K doesnot guarantee that for every group G of order k > K there exists a G -twin edge coloring of G , and see ourconcluding Section 5 for a further discussion concerning this issue.Surprisingly, in this paper we in fact provide an infinite family of connected graphs (which are trees)for which χ ′ g ( G ) ≥ ∆( G ) + 3 , see Theorem 3.4. Such phenomenon is not known for a few forefathersof this graph invariant discussed above (cf. additionally the conjecture in [ZLW02], and the best knownresult concerning this from [Hat05]), for which ∆( G ) + 2 labels are suspected to suffice for almostall connected graphs. In case of the group twin chromatic index, we conjecture that ∆( G ) + 3 labelsshould always be sufficient and confirm this for all trees (which are not isolated edges). On the way wealso discuss many rich families of trees for which such an upper bound can be improved. We then useour result concerning trees as a base case in a proof of a general upper bound for all graphs for whichthe parameter χ ′ g ( G ) is well defined. Namely, by means of a straightforward algorithmic construction(efficient for all connected graphs except possibly some trees) we finally provide a two-fold improvementof Theorem 1.4 of Johnston, whose proof is rather complex and lengthy. That is, we show that χ ′ g ( G ) ≤ G ) + col( G )) − for every graph G without isolated edges, where col( G ) denotes the coloringnumber of G (which is equal to the degeneracy of G plus ). This strengthens the thesis of Theorem 1.4,as col( G ) − ≤ ∆( G ) , while this inequality is sharp for many graph classes (e.g. for planar graphs, forwhich col( G ) ≤ whereas ∆( G ) is unbounded), and extends it towards colorings with elements of allAbelian groups, not just Z k . Assume G is an Abelian group of order n . The order of an element a = 0 is the smallest r such that ra = 0 . Recall that any group element ι ∈ G of order 2 (i.e., ι = 0 such that ι = 0 ) is called involution .It is well known by Lagrange Theorem that the order of any element of G divides |G| [Gal09]. Thereforeevery group of odd order has no involution. The fundamental theorem of finite Abelian groups statesthat a finite Abelian group G of order n can be expressed as the direct product of cyclic subgroups ofprime-power orders. This implies that G ∼ = Z p α × Z p α × . . . × Z p αkk where n = p α · p α · . . . · p α k k Sylwia Cichacz, Jakub Przybyło and p , p , . . . , p k are not necessarily distinct primes. This product is unique up to the order of the directproduct. When t is the number of these cyclic components whose order is a multiple of , then G has t − involutions. In particular every cyclic group of even order has exactly one involution.The sum of all the group elements is equal to the sum of the involutions and the neutral element. Thefollowing lemma was proved in [CNP04] (Lemma 8). Lemma 2.1 ([CNP04])
Let G be an Abelian group.1. If G has exactly one involution ι , then P g ∈G g = ι .2. If G has no involution, or more than one involution, then P g ∈G g = 0 . Anholcer and Cichacz proved a lemma about a partition of the set of all elements of G of order at most into two zero-sum sets (see [AC16], Lemma 2.4). Their result along with results proved by Cichacz (see[Cic17], Lemma 3.1) give the following lemma. Lemma 2.2 ([AC16, Cic17])
Let n , n , n be non-negative integers such that n + n + n = 2 k withinteger k ≥ , and k > if n n n = 0 . Let G be an Abelian group with involution set I ∗ = { ι , ι , . . . ,ι k − } and set I = I ∗ ∪ { } . Then there exists a partition A = { A , A , A } of I such that1. n = | A | , n = | A | , n = | A | ,2. P a ∈ A i a = 0 for i ∈ { , , } ,if and only if n , n , n
6∈ { , k − } . We start with the following lemmas:
Lemma 3.1
Let G be a finite Abelian group of order |G| ≥ having at most one involution. For anyelements a, b ∈ G such that b = 0 , a = b there exist elements x, y , x = y , { a, b } ∩ { x, y } = ∅ such that x + y = b − a . Proof:
For z ∈ G let S z/ = { t ∈ G : 2 t = z } . Observe that if there exist pairwise distinct t , t , t ∈ S z/ , then t − t and t − t are distinct involutions, therefore | S z/ | ≤ for any z ∈ G .We will show that there exists a desired solution of the equation x + y + a − b = 0 . Suppose first that
6∈ { a, b } . Then we may set x = b − a and y = 0 unless b = 2 a . In the latter case we must howeverhave |G| > (as b is the only involution in G ) and for any c ∈ G \ ( S − a/ ∪ { , b, a, − a } ) = ∅ , x = a + c and y = − c yield a solution. Similarly, if b = 0 and a = 0 , then for c ∈ G \ ( S b/ ∪ { , b } ) by setting x = b + c and y = − c we obtain a desired solution. Finally, if b = 0 , then x = c − a and y = − c are validprovided c ∈ G \ ( S a/ ∪ { , a, − a, a } ) , while to see that this last set is always nonempty it suffices tonote that if |G| = 6 and | S a/ | = 2 , then − a = 2 a . ✷ We also prove a somewhat stronger version of Lemma 2.2 (for the case when n = 0 ). Lemma 3.2
Let G be an Abelian group with involution set I ∗ = { ι , ι , . . . , ι k − } , k ≥ , and let I = I ∗ ∪ { } . Given an element ι ∈ I ∗ and positive integers n , n such that n + n = 2 k , n = 2 and n ≥ , there exists a partition A = { A , A } of I such thatroup twin edge coloring of graphs n = | A | , n = | A | ,2. P a ∈ A i a = 0 for i = 1 , ,3. ι A . Proof:
Recall that since I = { , ι , . . . , ι k − } is a subgroup of G , we have I ∼ = ( Z ) k . Observe that n and n have the same parity. If they are both even then there exists a partition A = { A , A } of I such that n = | A | , n = | A | and P a ∈ A i a = 0 for i = 1 , by Lemma 2.2. If now ι ∈ A , we aredone. If ι ∈ A , then for some ι ′ ∈ A there exists exactly one x ∈ I ∗ such that ι ′ + x = ι . Define now A ′ = { a + x, a ∈ A } and A ′ = { a + x, a ∈ A } . Note that P a ∈ A ′ i a = P a ∈ A i a + | A i | x = 0 for i = 1 , because | A i | is even. Hence { A ′ , A ′ } forms the desired partition.Assume now that n and n are both odd. One can see that the lemma holds for k = 2 . Suppose thatthe statement of the theorem is true for all groups with at least three and less than k − involutions. Letus establish it for groups with k − involutions. Let n ′ = n (mod 2 k − ) , n ′ = n (mod 2 k − ) . Let ι = ( i , i , . . . , i k ) and set ι ′ = ( i , i , . . . , i k − ) . If n > k − , then note that n = n ′ , n = n ′ + 2 k − and there exists a partition A ′ = { A ′ , A ′ } of ( Z ) k − such that n ′ = | A | ′ , n ′ = | A ′ | , P a ∈ A ′ i a = 0 for i = 1 , by Lemma 2.2. If now i k = 1 , then replace each element ( y , y , . . . , y k − ) of ( Z ) k − in any A ′ i by the element ( y , y , . . . , y k − , of ( Z ) k . Define A ′′ = { ( y, , y ∈ ( Z ) k − } and set A = A ′ , A = A ′ ∪ A ′′ . Suppose now i k = 0 . Note that then ι ′ = (0 , , . . . , . Replace each element ( y , y , . . . , y k − ) of ( Z ) k − in any A ′ i by the element ( y , y , . . . , y k − , except for ι ′ and some otherelement x ′ such that x ′ belongs to the same partition set as ι ′ (note we may do so, since ι ′ = 0 ); forthem we put ( ι ′ , and ( x ′ , . Define A ′′ = { ( y, , y ∈ ( Z ) k − , y / ∈ { ι ′ , x ′ }} and set A = A ′ and A = A ′ ∪ A ′′ ∪ { ( ι ′ , , ( x ′ , } . For n < k there exists a partition A ′ = { A ′ , A ′ } of ( Z ) k − suchthat ι ′ ∈ A ′ , n ′ = | A | ′ , n ′ = | A ′ | , P a ∈ A ′ i a = 0 for i = 1 , by the induction hypothesis. We thendefine A and A analogously as above. ✷ Lemma 3.3
Let T be a tree of order at least with maximum degree t and G be an Abelian group of oddorder k ≥ max { , t + 2 } . Then there exists a G -twin edge coloring of T , unless T is a (3 p − -regulartree and G ∼ = ( Z ) p for some integer p . Proof:
Let T be a tree with maximum degree t and G be an Abelian group of odd order k ≥ max { , t +2 } such that either T is not a t -regular tree or G 6∼ = ( Z ) p with p = t + 2 . Assume T is rooted at a vertex v with minimum degree t ′ ≥ in T . Let v , v , . . . , v m be all the remaining vertices of T which are notleaves, and denote their corresponding numbers of children by r , r , . . . , r m . Note that r i =deg ( v i ) − .Let N i = { v i w : w is a child of v i } for i = 0 , , . . . , m .If t ′ is even, take t ′ / pairs ( d , − d ) , . . . , ( d t ′ / , − d t ′ / ) of distinct elements of G and arbitrarily labelall edges incident with v with these. Then w ( v ) = 0 .Assume now t ′ is odd. Suppose first that t ′ < k − . Then in fact t ′ ≤ k − and there existnonzero pairwise distinct elements x, y, a ∈ G such that x + y + a = 0 by Lemma 3.1. Take x, y, a and ( t ′ − / distinct pairs ( d , − d ) , . . . , ( d ( t ′ − / , − d ( t ′ − / ) from the remaining elements of G andarbitrarily label all edges incident with v with these. Observe that w ( v ) = 0 then. Thus suppose now Sylwia Cichacz, Jakub Przybyło that t ′ = k − . Then we must have t ′ = t and by our choice of v , T must be a ( k − -regular tree.Therefore G 6∼ = ( Z ) p for all integers p , and hence there exist two distinct (nonzero) elements a and b suchthat a + b = − a . Then take , a, b and ( t − / pairs ( d , − d ) , . . . , ( d ( t − / , − d ( t − / ) from theremaining elements of G and label all edges incident with v with them. Observe that w ( v ) = a + b = − a then.Observe that for any edge e ∈ N we have f ( e ) = w ( v ) . In each next step now we will label edgesfrom the set N i only if the edge between v i and its parent v i is already labeled. We will do it in such away that f ( e ) / ∈ { f ( v i v i ) , } for any e ∈ N i and P e ∈ N i f ( e ) = 0 if r i > . Note that then any vertex v = v with deg ( v ) = 2 will have assigned a color equal to the label of the edge between v and its parent.Suppose first that r i is even. Then r i + 3 ≤ k and one can easily see that we can pick r i / pairs of alldistinct elements ( d , − d ) , . . . , ( d r i / , − d r i / ) from G not including f ( v i v i ) . We thus label the edgesof the set N i with these and we are done.Assume now r i is odd. Then r i + 4 ≤ k . Recall that f ( v i v i ) = 0 . For r i = 1 take any nonzeroelement g from G such that g / ∈ { f ( v i v i ) , w ( v i ) − f ( v i v i ) } (we can do this because |G| ≥ ) and labelthe edge from N i . Now, if r i ≥ , by Lemma 3.1 there exist two distinct nonzero elements x and y suchthat − f ( v i v i ) / ∈ { x, y } and x + y = − f ( v i v i ) . Take − f ( v i v i ) , x, y and ( r i − / pairs ( d , − d ) , . . . , ( d ( r i − / , − d ( r i − / ) from the remaining elements of G and arbitrarily label all edges from N i withthese. ✷ A G -twin edge coloring of a graph G is said to be nowhere-zero if it uses no label on any edge of G .Observe that by the proof above, if a tree T has even maximum degree at least , then T has a nowhere-zero G -twin edge coloring for any Abelian group G of odd order |G| ≥ ∆( T ) + 3 . Moreover, if T has oddmaximum degree at least , then it has a nowhere-zero G -twin edge coloring for any Abelian group G ofodd order |G| ≥ ∆( T ) + 2 , except for the case when T is a ( |G| − -regular tree.Thus we know that χ ′ g ( T ) ≤ ∆( T ) + 3 for all trees T of maximum degree at least except the regulartrees (for which χ ′ g ( T ) ≤ ∆( T ) + 4 holds). Before we show that such an upper bound is true for all trees,we present an infinite family of (regular) trees witnessing that this bound cannot be in general improved. Observation 3.4 If T is a regular tree of order at least such that ∆( T ) = 3 p +1 − for some positiveinteger p , then χ ′ g ( T ) ≥ ∆( T ) + 3 . Proof:
Observe that ∆( T ) = 3 p +1 − ≡ . Therefore χ ′ g ( T ) ≥ ∆( T ) + 2 by Theorem 1.3.Assume, to the contrary, that χ ′ g ( T ) = ∆( T ) + 2 . Let f be a G -twin edge coloring of T with G =( Z ) p +1 , and let v be an internal vertex of T , i.e. such that deg ( v ) = ∆( T ) . Then there are exactly twoelements a, b ∈ ( Z ) p +1 that are not assigned to any edge incident with v . Since w ( v ) = − ( a + b ) andit is impossible in ( Z ) p +1 that − ( a + b ) = a or − ( a + b ) = b , we obtain that w ( v ) = c = f ( v v ) forsome v ∈ N ( v ) , c / ∈ { a, b } . If v is a leaf of T , then w ( v ) = w ( v ) , a contradiction. Thus v is not aleaf of T and so deg ( v ) = ∆( T ) . Analogously as above, w ( v ) = c = f ( v v ) for some v ∈ N ( v ) ,and since f distinguishes v and v by their corresponding sums, then c = c , and hence v = v . If v is a leaf of T , we obtain a contradiction. Otherwise we continue this process, and since T has no cyclesand is finite, we eventually must reach a leaf v r such that v r ∈ N ( v r − ) and w ( v r ) = c r = w ( v r − ) , acontradiction. ✷ roup twin edge coloring of graphs Lemma 3.5
Let T be a tree with maximum degree t ≥ such that any vertex of degree has at least oneneighbor of degree and let G be an Abelian group with exactly one involution ι , |G| ≥ t + 2 . Then thereexists a G -twin edge coloring of T . Proof:
Set k = |G| . We will define a G -twin edge coloring f : E ( T ) → G .Let T be rooted at a vertex v of degree t . Let v , v , . . . , v m be all the remaining vertices of T whichare not leaves, and denote their corresponding numbers of children by r , r , . . . , r m . Set N i = { v i w : w is a child of v i } for i = 0 , , . . . , m .If t is even take t/ distinct pairs ( d , − d ) , . . . , ( d t/ , − d t/ ) of the elements of G and arbitrarily labelall edges incident with v with them. Then w ( v ) = 0 . If t is odd, then k > t + 2 ≥ . Therefore, byLemma 3.1, in the group G we have nonzero elements a = b such that a + b = ι . Take , a, b and ( t − / distinct pairs ( d , − d ) , . . . , ( d ( t − / , − d ( t − / ) from the remaining elements of G and label all edgesincident with v with them. Observe that v is assigned a color w ( v ) = a + b = ι then.The main idea of the proof is similar to the proof of Lemma 3.3 – in each next step we will label edgesfrom the set N i only if the edge between v i and its parent (say v i ) is already labeled. This time howeverthe label is allowed for an edge, but only for edges belonging to a set N i with r i > .Suppose first that r i is even. Then r i + 4 ≤ k and one can easily see that we can pick r i / distinctpairs ( d , − d ) , . . . , ( d r i / , − d r i / ) of elements of G not including f ( v i v i ) . We label the edges in N i with these and we are done.Assume now r i is odd, thus r i + 3 ≤ k . For r i = 1 , take any nonzero element g from G such that g / ∈ { f ( v i v i ) , w ( v i ) − f ( v i v i ) } (we can do this because |G| ≥ ) and label the edge from N i . Assume nowthat r i ≥ . If f ( v i v i ) = 0 , take and ( r i − / distinct pairs ( d , − d ) , . . . , ( d ( r i − / , − d ( r i − / ) from elements of G that are different from f ( v i v i ) , − f ( v i v i ) and arbitrarily label the edges of the set N i with these elements. Suppose then that f ( v i v i ) = 0 . By Lemma 3.1, there exist distinct nonzeroelements x, y ∈ G such that x + y = ι . Thus take ι, x, y and ( r i − / distinct pairs ( d , − d ) , . . . , ( d ( r i − / , − d ( r i − / ) from the remaining elements of G and label all edges from N i with them. ✷ The famous Catalan-Mihˇailescu Theorem says that the only solution in the natural numbers of theequation x a − y b = 1 for a, b > , x, y > is x = 3 , a = 2 , y = 2 , b = 3 [Mih04]. Therefore χ ′ g ( K , p − ) ≤ p − for p ≥ by Lemma 3.3. However the tree K , p − does not have a ( Z ) p -twin edge coloring. Indeed, for suppose we are able to label K , p − appropriately with elements from ( Z ) p . In such a situation we would have to use p − distinct elements of ( Z ) p on the edges, whichwould leave us three distinct elements, g , g , g unused. The weighted degree of the central vertex wouldbe − ( g + g + g ) . This should be distinct from all other weighted degrees, so one of the equalities − ( g + g + g ) = g , − ( g + g + g ) = g or − ( g + g + g ) = g would have to be satisfied. Inall cases it follows that g i = g j for i = j , i, j ∈ { , , } , a contradiction. Moreover we could extend thisarguments similarly as in the proof of Observation 3.4 to the case of any (2 p − -regular tree. However,for a group G having more than one involution we are able to prove the following. Lemma 3.6
Let T be a tree with maximum degree t ≥ such that any vertex of degree has at least oneneighbor of degree and let G be an Abelian group of order k ≥ t + 2 with more than one involution. Let p be an integer such that p ∈ { log ( t + 2) , log ( t + 3) } . If G 6∼ = ( Z ) p , then there exists a G -twin edgecoloring of T . Sylwia Cichacz, Jakub Przybyło
Proof:
Let G be an Abelian group of order k with involution set I ∗ = { ι , ι , . . . , ι p − } , p > . Let I = I ∗ ∪ { } . Note that G has even order. Since the group G can be expressed as the direct product ofcyclic subgroups of prime-power orders one can easily see that either k = 2 p or k > p +1 − . We willdefine a G -twin edge coloring f : E ( T ) → G .As before let T be a rooted tree with root v such that deg( v ) = ∆( T ) = t . Let v , v , . . . , v m be allthe remaining vertices of T which are not leaves, and denote their corresponding numbers of children by r , r , . . . , r m . Let N i = { v i w : w is a child of v i } for i = 0 , , . . . , m .If t is even and k = t + 2 , then from the assumption that G 6∼ = ( Z ) p we deduce that t > p +1 − .Take ι , ι , . . . , ι p − and ( t + 2 − p ) / distinct pairs ( d , − d ) , . . . , ( d ( t +2 − p ) / , − d ( t +2 − p ) / ) fromelements of G and arbitrarily label all edges incident with v with these. Observe that then w ( v ) = ι by Lemma 2.1. Suppose t is even and k ≥ t + 4 . If now t < p − then there exists a partition A = { A , A , A } of I such that | A | = t ≥ , | A | = 1 , | A | = 2 p − − t and P a ∈ A i a = 0 for i ∈ { , , } by Lemma 2.2. Note that / ∈ A . Label all edges incident with v with the elementsof A , hence w ( v ) = 0 . If now t ≥ p − then obviously k > p +1 − . Thus by Lemma 2.2,there exists a partition A = { A , A , A } of I such that | A | = 2 p − , | A | = 1 , | A | = 3 and P a ∈ A i a = 0 for i ∈ { , , } . We take elements from A and ( t + 4 − p ) / distinct pairs ( d , − d ) ,. . . , ( d ( t +4 − p ) / , − d ( t +4 − p ) / ) from the elements of G and label all edges incident with v with these.Observe that then w ( v ) = 0 .Assume now that t is odd. If t ≥ p − , then take ι , ι , . . . , ι p − and ( t + 1 − p ) / distinctpairs ( d , − d ) , . . . , ( d ( t +1 − p ) / , − d ( t +1 − p ) / ) from the elements of G and arbitrarily label all edgesincident with v with them. Observe that then w ( v ) = 0 . If t < p − then there exists a partition A = { A , A , A } of I such that | A | = t ≥ , | A | = 1 , | A | = 2 p − − t ≥ and P a ∈ A i a = 0 for i ∈ { , , } by Lemma 2.2. Label all edges incident with v by the elements of A , thus w ( v ) = 0 .Finally suppose that p − t . By the assumption G 6∼ = ( Z ) p , this implies that k > p . Thusby Lemma 2.2 there exists a partition A = { A , A , A } of I such that | A | = t − ≥ , | A | = 1 , | A | = 2 p + 1 − t and P a ∈ A i a = 0 for i ∈ { , , } . Take elements from A and one pair ( d , − d ) fromthe elements of G and label all edges incident with v with these elements. Observe that then w ( v ) = 0 .The main idea of the further part of the proof is the same as in the proof of Lemma 3.5. In each nextstep we will label edges from the set N i only if the edge between v i and its parent (say v i ) is alreadylabeled.Suppose first that r i is odd, thus r i +3 ≤ k . If r i = 1 then we are taking any non-zero element g from G such that g / ∈ { f ( v i v i ) , w ( v i ) − f ( v i v i ) } (we can do this because |G| ≥ ) and label the edge in N i withit. Let r i ≥ . Suppose that f ( v i v i ) / ∈ I ∗ . Then for r i < p − there exists a partition A = { A , A , A } of I such that | A | = r i , | A | = 1 , | A | = 2 p − − r i and P a ∈ A j a = 0 for j ∈ { , , } by Lemma 2.2.Label all edges from N i by the elements of A . If p − r i , then t ≥ k + 2 and G 6∼ = ( Z ) p imply that k > p and r i ≥ . Therefore there exists a partition A = { A , A , A } of I such that | A | = r i − , | A | = 1 , | A | = 2 p + 1 − r i and P a ∈ A j a = 0 for j ∈ { , , } by Lemma 2.2. Take elements from A and one pair ( d , − d ) such that f ( v i v i ) / ∈ { d , − d } from the elements of G and label all edgesfrom N i with these. For r i ≥ p − take ι , ι , . . . , ι p − and ( r i + 1 − p ) / pairs ( d , − d ) , . . . , ( d ( r i +1 − p ) / , − d ( r i +1 − p ) / ) from the elements of G that are different from f ( v i v i ) and label the edgesof the set N i with them. Assume now f ( v i v i ) ∈ I ∗ . Then for r i < p − there exists a partition A = { A , A } of I such that | A | = r i , | A | = 2 p − r i , P a ∈ A j a = 0 for j ∈ { , } and f ( v i v i ) / ∈ A roup twin edge coloring of graphs N i by the elements of A then. For r i ≥ p − there exists apartition A = { A , A } of I such that | A | = 2 p − , | A | = 3 , P a ∈ A j a = 0 for j ∈ { , } and f ( v i v i ) / ∈ A by Lemma 3.2. Recall that for p − ≤ r i we have k > p +1 − . Label all edges from N i by the elements of A and ( r i + 3 − p ) / distinct pairs ( d , − d ) , . . . , ( d ( r i +3 − p ) / , − d ( r i +3 − p ) / ) from the elements of G .Assume that r i is even, hence k ≥ r i + 4 . If r i = 2 and G 6∼ = ( Z ) p , take any ( d , − d ) such that f ( v i v i ) / ∈ { d , − d } to label the two edges in N i and we are done. For G ∼ = ( Z ) p one can see that weare able to take g , g ∈ G such that g = g , f ( v i v i )
6∈ { g , g } and g + g = w ( v i ) − f ( v i v i ) . Labelthe edges from N i by g and g . Let r i ≥ . Assume first that r i ≥ p . If f ( v i v i ) / ∈ I , take all elementsof I and ( r i − p ) / distinct pairs ( d , − d ) , . . . , ( d ( r i − p ) / , − d ( r i − p ) / ) from the elements of G notincluding f ( v i v i ) and label all edges from N i with them. If on the other hand f ( v i v i ) ∈ I then thereexists a partition A = { A , A } of I such that | A | = 2 p − , | A | = 4 , P a ∈ A j a = 0 for j ∈ { , } and f ( v i v i ) / ∈ A (it is trivial for p = 2 , while otherwise: if f ( v i v i ) ∈ I ∗ it follows directly by Lemma 3.2,and if f ( v i v i ) = 0 then it is sufficient to apply Lemma 2.2 to obtain appropriate sets A ′ , A ′ , A ′ with | A ′ | = 2 p − , | A ′ | = 1 , | A ′ | = 3 and set A = A ′ , A = A ′ ∪ A ′ ). We take elements from A and ( r i + 4 − p ) / distinct pairs ( d , − d ) , . . . , ( d ( r i +4 − p ) / , − d ( r i +4 − p ) / ) from the elements of G andlabel all edges from N i with them. Now if r i = 2 p − , then k ≥ p +1 , and hence we may take r i / distinct pairs ( d , − d ) , . . . , ( d r i / , − d r i / ) from the elements of G not including f ( v i v i ) and label alledges from N i with these. Finally, if r i ≤ p − , then similarly as above, by Lemma 3.2 or 2.2 thereexists a partition A = { A , A } of I such that | A | = r i , | A | = 2 p − r i , P a ∈ A j a = 0 for j ∈ { , } and f ( v i v i ) / ∈ A . We then label the edges in N i with all elements from A . ✷ Observe that if T is a tree with maximum degree ∆( T ) = 2 p − ≥ such that any vertex of degree has at least one neighbor of degree that is not isomorphic to a (2 p − -regular tree then using exactlythe same method as in the proof of Lemma 3.6 for the root of degree r
6∈ { , p − } we obtain that T hasa ( Z ) p -twin edge coloring.By Theorem 1.2 and Lemmas 3.5 and 3.6 we deduce the following. Observation 3.7
Let T be a tree with even maximum degree such that any vertex of degree has at leastone neighbor of degree . If ∆( T ) = 2 p − for every integer p , then χ ′ g ( T ) ≤ ∆( T ) + 2 . ✷ Finally, we obtain the following upper bound, which is tight according to Observation 3.4.
Theorem 3.8 If T is a tree of order at least then χ ′ g ( T ) ≤ ∆( T ) + 2 if ∆( T ) is odd and T is not (3 p − -regular for some integer p ≥ and χ ′ g ( T ) ≤ ∆( T ) + 3 otherwise. Proof:
Observe that for k ∈ { , , , } all groups of order k are isomorphic to Z k , and therefore everytree T having maximum degree at most has χ ′ g ( T ) ≤ ∆( T ) + 2 by Theorem 1.2. Thus we may assumethat ∆( T ) ≥ . For T not being regular we are then done by Lemma 3.3. Assume now T is a regulartree. If ∆( T ) = 3 p − for every integer p then we are again done by Lemma 3.3. Suppose then that ∆( T ) = 3 p − and G is a group of order p + 1 . Then ∆( T ) + 3 is even but cannot be equal to r for anynatural number r by the Catalan-Mihˇailescu Theorem [Mih04], and therefore the existence of a G -twinedge coloring of T follows by Lemmas 3.5 and 3.6. ✷ Sylwia Cichacz, Jakub Przybyło
Recall that for a given graph G by col( G ) we denote its coloring number, that is the least integer k suchthat each subgraph of G has minimum degree less than k . Equivalently, it is the smallest k for whichwe may linearly order all vertices of G into a sequence v , v , . . . , v n so that every vertex v i has at most k − neighbors preceding it in the sequence. Hence col( G ) ≤ ∆( G ) + 1 . Note that col( G ) equals thedegeneracy of G plus , and thus the result below may be formulated in terms of either of the two graphinvariants. Theorem 4.1 If G is a connected graph of order at least then χ ′ g ( G ) ≤ G ) + col( G )) − . Proof:
Suppose first that col( G ) = 2 . For ∆( G ) = 2 , the statement of the theorem is true by Theorem 1.2,while for ∆( G ) ≥ it follows from Theorem 3.8.So we may assume that col( G ) ≥ . Fix any Abelian group G of order |G| ≥ G ) + col( G )) − .Let v , v , . . . , v n be the ordering of V ( G ) witnessing the value of col( G ) . We will label the edges of G with elements of G in n − stages, each corresponding to a consecutive vertex from among v , v , . . . , v n .Initially no edge is labeled. Then at each stage i , i = 2 , , . . . , n , we label all backward edges of v i , i.e.every edge v j v i ∈ E with j < i ; such a vertex v j is called a backward neighbor of v i . We will chooselabels avoiding (most of the) sum conflicts between already analyzed vertices and so that at all times thepartial edge coloring obtained is proper. To this end we will make sure that at the end of every stage i ,the conditions ( ◦ )-( ◦ ) below hold. Let I i denote the set of indices j of all vertices v j in { v , v , . . . , v i } each of which has a neighbor v k of degree in G with k > i (note that for any i , if an index j ∈ { , . . . , i } does not belong to I i , then j does not belong to any set I t with t ≥ i ). By w ( v t ) , for each t in { , . . . , n } ,we mean the contemporary sum at v t (with unlabeled edges contributing to such a sum):( ◦ ) adjacent edges must be labeled differently;( ◦ ) for every j ∈ I i such that v j has a neighbor in { v , . . . , v i } : w ( v j ) = 0 ;( ◦ ) for every edge v j v k ∈ E ( G ) such that j, k / ∈ I i , j < k ≤ i and v k has at least neighbors in { v , . . . , v i } or v j has a neighbor in { v k +1 , . . . , v i } : w ( v j ) = w ( v k ) .Note that if we are able to assure ( ◦ )-( ◦ ) to hold after every stage, then the edge coloring of G obtainedat the end of our construction will be proper (by ( ◦ )), and moreover the neighbors will be distinguishedby their corresponding sums, as desired. To see the latter of these, i.e. that w ( v j ) = w ( v k ) for everyedge v j v k ∈ E ( G ) with ≤ j < k ≤ n , consider first the case when deg( v k ) ≥ . Then the factthat w ( v j ) = w ( v k ) follows directly from ( ◦ ), as I i = ∅ by definition for i = n . Assume thus that deg( v k ) = 1 , and hence deg( v j ) ≥ . Denote by v t the neighbor of v j in G with the largest index t (hence t ≥ k ). Then if t > k , by ( ◦ ) we must have had w ( v j ) = w ( v k ) after stage t and this couldnot change in the further part of the construction (as no other unlabeled edges incident with v j or v k areleft after stage t ). If finally t = k , by ( ◦ ) we must have had w ( v j ) = 0 = w ( v k ) after stage k − andthe inequality w ( v j ) = w ( v k ) could not be violated regardless of the choice of the label for v j v k in thefollowing stage (nor in any further ones). In order to prove the theorem it is thus indeed sufficient to showthat we are able to satisfy ( ◦ )-( ◦ ) after every stage using labels in G .So assume we are about to perform step i of the construction for some i ∈ { , . . . , n } and thus farall our requirements have been fulfilled. Let v i , v i , . . . , v i b be all backward neighbors of v i (hence b ≤ col( G ) − ), and set b = 0 if there are none. If b > , subsequently for j = 1 , , . . . , b − (if roup twin edge coloring of graphs b > ) we choose weights for v i j v i consistently with our rules. Thus by ( ◦ ) we cannot use at most (∆( G ) −
1) + ( j − ≤ ∆( G ) + col( G ) − labels of already colored edges adjacent with v i j v i to labelit. If i j ∈ I i , in order to obey ( ◦ ) we cannot use at most one more label for v i j v i so that w ( v i j ) = 0 afterwards, while if i j / ∈ I i we will choose the label for v i j v i so that afterwards the sum at v i j is distinctfrom the sums of all its neighbors except possibly v i – this blocks at most ∆( G ) − additional labels for v i j v i . As we have thus altogether at most ∆( G ) + col( G ) − G ) − forbidden labels, we are leftwith at least col( G ) ≥ available options in G to label v i j v i . We choose any of these, except for the casewhen j = b − , when we make the choice so that w ( v i b ) = w ( v i ) afterwards (note that though the choiceof label for v i b − v i is performed before the one for v i b v i , the latter one counts in the the sums of both, v i b and v i , and thus the label of v i b v i does not influence the distinction of w ( v i b ) from w ( v i ) ). Next, for v i b v i we analogously as above cannot use at most ∆( G ) + col( G ) − labels by ( ◦ ). Moreover, again atmost additional label might be blocked for v i b v i by ( ◦ ) if i b ∈ I i or at most ∆( G ) − labels otherwise– so that the obtained sum at v i b is distinct from the sums of its neighbors except possibly v i . Similarly atmost additional label might be blocked for v i b v i by ( ◦ ) if i ∈ I i or at most col( G ) − labels otherwise– so that the obtained sum at v i is distinct from the sums of its backward neighbors except possibly v i b .Altogether at most (∆( G ) + col( G ) −
3) + (∆( G ) −
1) + (col( G ) − < |G| labels are thus forbidden,and hence we have at least one label available for v i b v i .By such a construction it is clear that conditions ( ◦ ) and ( ◦ ) hold after stage i . As for condition ( ◦ )it is also straightforward that it holds for every edge v j v k ∈ E ( G ) such that j, k / ∈ I i , j < k ≤ i exceptpossibly v i b v i , for which we were admitting a possible conflict w ( v i b ) = w ( v i ) . By our constructionthis however could only happen if b = 1 , as for b ≥ we prevented this by our choice of the label of v i b − v i . Then however v i b v i does not meet the assumptions of ( ◦ ) after stage i , so we need not have w ( v i b ) = w ( v i ) according to our rules.As discussed earlier, after step n of the construction we obtain a desired edge labeling of G . ✷ By Theorem 4.1 we in particular obtain that the following is true.
Corollary 5.1 If G is a connected planar graph of order at least then χ ′ g ( G ) ≤ G ) + 7 . We believe however that a stronger upper bound should hold even for all graphs, and we pose the followingconjecture.
Conjecture 5.2 If G is a connected graph of order at least then χ ′ g ( G ) ≤ ∆( G ) + 3 . By Observation 3.4 this could not be improved. In this context it would also be interesting to settle forwhich trees T we actually have the equality χ ′ g ( T ) = ∆( T ) + 3 .At the end notify that so far only for only one family of trees (namely (3 p − -regular trees with p ≥ )we can conclude that there exists a G -twin edge coloring for any Abelian group G of order k ≥ χ ′ g ( G ) .Recall that the fact that χ ′ g ( G ) ≤ K for a given graph G and a constant K does not guarantee that forevery group G of order k > K there exists a G -twin edge coloring of G . We may almost guarantee this inthe case of trees though. Note that for an Abelian group G , |G| ≥ , with at most one involution we canimprove Lemma 3.1 so that x, y are additionally nonzero. Moreover for k ≥ and n ≥ in Lemma 3.2we can require that also / ∈ A . Hence using a similar method as in the proofs of Lemmas 3.3, 3.5 and 3.6one can show that for any Abelian group G of order k > ∆( T )+ 3 . With a bit of extra effort we thus could2 Sylwia Cichacz, Jakub Przybyło obtain upper bounds of similar flavor as the ones in Theorem 3.8 for all forests. These we can howeverderive effortlessly from our results for the case of labelings with cyclic groups Z k . Thus we conclude ourpaper with Theorem 5.3 below, containing exactly these upper bounds for the twin chromatic index offorests.One can easily see that a path of order at least has a Z k -twin edge labeling for any odd k ≥ . Thenby Theorem 1.2 and Lemma 3.3 we directly obtain the following. Theorem 5.3
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