# Regular bipartite graphs are antimagic

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1 Regular bipartite graphs are antimagic Daniel W. Cranston Abstract A labeling of a graph G is a bijection from E(G) to the set {1, 2,..., E(G) }. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than K 2 is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem. 1 Introduction In this paper, we study a problem of edge-labeling. For convenience, we formally define a labeling of a graph G to be a bijection from E(G) to the set {1,..., E(G) }. A vertexsum for a labeling is the sum of the labels on edges incident to a vertex v; we also call this the sum at v. A labeling is antimagic if the vertex-sums are pairwise distinct. A graph is antimagic if it has an antimagic labeling. Hartsfield and Ringel [2] introduced antimagic labelings in 1990 and conjectured that every connected graph other than K 2 is antimagic. The most significant progress on this problem is a result of Alon, Kaplan, Lev, Roditty, and Yuster [1], which states the existence of a constant c such that if G is an n-vertex graph with δ(g) c log n, then G is antimagic. Large degrees satisfy a natural intuition: the more edges that are present, the more flexibility there is to arrange the labels and possibly obtain an antimagic labeling. For proofs of some other special cases, see [3]. Alon et al. also proved that G is antimagic when (G) V (G) 2, and they proved that all complete multipartite graphs (other than K 2 ) are antimagic. Hartsfield and Ringel proved that paths, cycles, wheels, and complete graphs are antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our proof relies heavily on the Marriage Theorem, which states that every regular bipartite graph has a 1-factor (see [4], page 111). By induction on the vertex degree, it follows that a regular bipartite graph decomposes into 1-factors. Recall that a k-factor is a k-regular spanning subgraph, so the union of any k 1-factors from such a decomposition is a k-factor. Throughout this paper, we refer to the partite sets of the given bipartite graph as A and B, each having size n. University of Illinois, Urbana, 1

2 With respect to a given labeling, two vertices conflict if they have the same sum. We view the process of constructing an antimagic labeling as resolving the potential conflict for every pair of vertices. We will label the edges in phases. When we have labeled a subset of the edges, we call the resulting sum at each vertex a partial sum. Our general approach is to label all but a single 1-factor so that the partial sums in A are multiples of 3, while the partial sums in B are non-multiples of 3. At this stage no vertex of A conflicts with a vertex of B. We then label the final 1-factor with reserved labels that are multiples of 3 so that we resolve all potential conflicts within A and within B. Before we begin the general approach, we observe two facts that together show that 2-regular graphs are antimagic. Fact 1. [2] Every cycle is antimagic. Proof: Assign the labels to edges as 1, 3,..., n, n 1,..., 4, 2 in order around an n- cycle (if n is odd; otherwise, n and n 1 are switched in the middle). The sums are 4, 8,..., 2n 6, 2n 2, 2n 1, 2n 4,..., 10, 6, 3; that is, the sums of consecutive odd integers are even multiples of 2, while the sums of consecutive even integers are odd multiples of 2 (the sums 3 and 2n 1 also appear). Fact 2. If G 1 and G 2 are each regular antimagic graphs, then the disjoint union of G 1 and G 2 is also antimagic. Proof: Index G 1 and G 2 so that vertices in G 2 have degree at least as large as those in G 1. Let m 1 = E(G 1 ). Label G 1 with an antimagic labeling, using the first m 1 labels. Label G 2 by adding m 1 to each label in an antimagic labeling of G 2. Let k be the degree of the vertices in G 2. Adding m 1 to each label in G 2 adds m 1 k to the sum at each vertex of G 2, so the new labeling of G 2 has distinct vertex-sums. Hence there are no conflicts within G 1 and no conflicts within G 2. There are also no conflicts between a vertex in G 1 and one in G 2, since each vertex-sum in G 1 is less than m 1 k and each vertex-sum in G 2 is greater than m 1 k. More generally, given any labeling of a regular graph, adding the same amount to each label does not change the pairs of vertices that conflict. Fact 1 and Fact 2 immediately yield: Corollary 3. Every simple 2-regular graph is antimagic. For degrees larger than 2, we will consider odd and even degree separately. Although 2-regular graphs are easy, the general construction is a bit more complicated for even degree than for odd degree. 2 Regular bipartite graphs with odd degree We have observed that a k-regular bipartite graph G decomposes into 1-factors. We can combine these 1-factors in any desired fashion. In particular, when k is odd and 2

3 at least 5, we can decompose G into a (2l + 2)-factor and a 3-factor, where l 0. Our aim will be to combine special labelings of these two factors to obtain an antimagic labeling of G. The case k = 3 is handled separately; we do this before the general argument. Theorem 4. Every 3-regular bipartite graph is antimagic. Proof: Since G has 3n edges, we have the same number of labels in each congruence class (mod 3). For convenience, we use the term j-labels to designate the first n positive integers that are congruent to j (mod 3), where j {0, 1, 2}. Decompose G into a 1-factor H 1 and a 2-factor H 2. We will reserve the 0-labels for H 1. We will label H 2 with the 1-labels and 2-labels so that the partial sum at each vertex of A is 3n. We do this by pairing each 1-label i with the 2-label 3n i. These pairs have sum 3n; at each vertex of A, we use the two integers in some pair. Subsequently, every assignment of 0-labels to H 1 yields distinct vertex-sums within A. We have assigned a pair of labels at each vertex of A in H 2, but we have not decided which edge gets which label. Next we try to make this choice so that in H 2 the partial sums at vertices of B will not be multiples of 3. In each component of H 2, we will fail at most once. Let C be a cycle that is a component of H 2. We have a 1-label and a 2-label at each vertex of A. As we follow C, if we have a 1-label and then a 2-label at a vertex of A, then the next vertex of A should have a 2-label followed by a 1-label (and vice versa), since the sum of two 1-labels or two 2-labels is not a multiple of 3. If V (C) A is even, then we succeed throughout; if V (C) A is odd, then at one vertex of C in B we will have a 1-label and a 2-label. Call such a vertex of B bad. A cycle in H 2 has a bad vertex only if it has length at least 6, so at most n/3 vertices in B will be bad. Let m be the number of bad vertices. To avoid conflicts between vertices of A and bad vertices of B, we will make the vertex-sum at each bad vertex smaller than at any vertex of A. Furthermore, we will make the partial sums in H 2 at these vertices equal. Consider the 1-labels and 2-labels from 1 through 3m 1; group them into pairs j and 3m j. The sum in each such pair is 3m, which is at most n. Allocate the pairs for H 2 to vertices of A so that at each bad vertex of B, the labels are the small elements from pairs in the original pairing and form a pair with sum 3m in this most recent pairing. Now we need to label H 1. We must achieve three goals: resolve all conflicts among the good vertices in B, resolve all conflicts among the bad vertices in B, and resolve all conflicts between A and the bad vertices in B. We consider the last goal first. For every assignment of 0-labels to H 1, the vertexsums in A will be {3n + 3, 3n + 6,..., 6n 3, 6n}. To ensure that the vertex-sums at the bad vertices in B will be less than 3n + 3, we use the smallest 0-labels at the bad vertices. Since there are at most n/3 bad vertices, every 0-label at such a vertex is at most n. Thus, every sum at a bad vertex is at most 2n, which is less than 3n. Furthermore, the sums at bad vertices are 3m plus distinct 0-labels; hence they are distinct, which completes the second goal. For the first goal, let b 1, b 2, b 3,... denote the good vertices of B in order of increasing partial sum from H 2 (there may be ties). We assign the remaining 0-labels to edges 3

4 of H 1 at b 1, b 2,... in increasing order. Since the 0-labels are distinct, this prevents conflicts among the good vertices in B. For larger odd degree, we will construct an antimagic labeling from special labelings of two subgraphs. Like the labeling we constructed for 3-regular graphs, the first labeling will have equal sums at vertices of A, but this time we guarantee that all sums at vertices of B are not congruent (mod 3) to the sums at vertices of A. Lemma 5. If G is a (2l + 2)-regular bipartite graph with parts A and B of size n, then G has a labeling such that the sum at each vertex of A is some fixed value t and the sum at each vertex of B is not congruent to t(mod 3). Proof: As remarked in the introduction, we can decompose G into a 2l-factor H 2l and a 2-factor H 2. Let m = (2l + 2)n; thus m is the largest label. Since m is even, we can partition the labels 1 through m into pairs that sum to m + 1. Choose a so that m + 1 2a(mod3); each pair consists either of two elements in the same congruence class as a(mod 3) or of elements in the two other congruence classes (mod 3). Call these like-pairs and split-pairs, respectively. At each vertex of A, we will use l of these pairs as labels in H 2l. Thus each vertex of A will have partial sum (m + 1)l in H 2l ; we will assign the pairs so that the partial sums in B are not congruent to (m+1)l( mod 3). We use the pairs in which the smaller label ranges from 1 to ln. Note that H 2l decomposes into even cycles (for example, we can take 2l 1-factors two at a time to generate 2-factors whose union is H 2l ). For each cycle in the decomposition of H 2l into even cycles, at vertices of A we use pairs of labels of the same type: all like-pairs or all split-pairs. When using split-pairs, we assign the labels so that the same congruence class (mod 3) is always first. If we have all like-pairs or all split-pairs, this ensures that at each vertex of B, each cycle contributes an amount to the sum that is congruent to 2a(mod 3). There is at most one cycle where we are forced to use both like-pairs and split-pairs. Let x and y be the vertices of B where, in this cycle, we switch between like-pairs and split-pairs. At each vertex of A, the partial sum in H 2l is (m + 1)l. At each vertex of B, except x and y, the partial sum is congruent to (m + 1)l(mod 3). On H 2, we use the remaining pairs of labels so that we add m + 1 to each partial sum in A, but what we add to each partial sum in B is not congruent to m+1( mod 3). If we can do this (and treat x and y specially), then the sum at each vertex of A will be (m + 1)(l + 1), while at each vertex of B the sum will be in a different congruence class (mod 3) from (m + 1)(l + 1). On each cycle, we use the pairs of labels that contain the smallest unused labels. Thus, every third pair we use is a like-pair; the others are split pairs. We begin with a like-pair and alternate using a like-pair and a split-pair until the like-pairs allotted to that cycle are exhausted. For the remaining split-pairs, we alternate them in the form (a + 1, a + 2) followed by (a + 2, a + 1); in this way the sum of the two labels used at any vertex of B is not congruent to 2a(mod 3). If no like-pair is available to be used on the cycle, then the cycle has length 4 and we label it with split-pairs in the form (a + 1, a + 2), (a + 2, a + 1), and the same property holds. One or two cycles in H 2 may contain the vertices x and y, where the sum in H 2l differs by 1 from a value congruent to (m + 1)l(mod 3). Suppose that the sums in H 2l 4

5 at x and y are (m + 1)l + t 1 and (m + 1)l + t 2. We want the sum at x in H 2 to be either 2a t 1 + 1(mod3) or 2a t 1 + 2(mod3). Similarly, we want the sum at y in H 2 to be in {2a t 2 + 1, 2a t 2 + 2}. The more difficult case is when x and y lie on the same cycle in H 2. However, given the realization that we have two choices each for the sums (mod 3) at x and y, it is not difficult to adapt the labeling given above for cycles of H 2 so that it applies in the current case as well. At these vertices we want the contribution from H 2 to be congruent to 2a(mod3). We deal with these first and can then make the argument above for the remaining cycles. If x and y lie on a single 4-cycle, then we use two like-pairs or two split-pairs ordered as (a + 1, a + 2), (a + 1, a + 2). If one or both of x and y lie on a longer cycle, then at each we put edges from two like-pairs or from two split-pairs ordered as (a + 1, a + 2), (a + 1, a + 2). The remaining pairs, whether they are like-pairs or split-pairs as we allocate them to this cycle, can be filled in so that like-pairs are not consecutive anywhere else and neighboring split-pairs alternate their orientation. Thus the labeling of H 2 enables us to keep the overall sum at each vertex of B out of the congruence class of (m + 1)(l + 1)(mod3). Lemma 6. If G is a 3-regular bipartite graph with parts A and B, where B = {b 1,..., b n }, then G has a labeling with the numbers {1, 2, 3,..., 3n} so that at each b i the sum is 3n + 3i, and for each i exactly one vertex in A has sum 3n + 3i. Proof: Decompose G into three 1-factors: R, S, and T. In R, use label 3i 2 on the edge incident to b i ; let a i be the other endpoint of this edge. In S, use label 3n + 3 3i on the edge incident to a i ; call the other endpoint of this edge b i. In T, use label 3i 1 on the edge incident to b i ; call the other endpoint of this edge a i. Note that each 1-factor received the labels from one congruence class (mod3). The partial sum in S T at each vertex of B is 3n + 2. Hence, the sum at each vertex b i is 3n + 3i. Similarly, the partial sum in R S at each vertex of A is 3n + 1. Hence, the vertex-sum at each a i is 3n + 3i. Theorem 7. Every regular bipartite graph of odd degree is antimagic. Proof: Let G be a regular bipartite graph of degree k. Theorem 4 is the case k = 3. For k > 3, let k = 2l + 5 with l 0, and decompose the graph G into a 3-factor H 3 and a (2l + 2)-factor H 2l+2. Label H 2l+2 as in Lemma 5; this uses labels 1 through (2l + 2)n. Add 3n to each label, leaving labels 1 through 3n for H 3. Each vertex-sum increases by 3n(2l + 2), which is a multiple of 3, so the congruence properties obtained in Lemma 5 remain true for the new labeling. Let b i denote the vertices of B in order of increasing partial sum in H 2l+2. Label H 3 as in Lemma 6. Because all the partial sums in H 3 are multiples of 3, the labeling of H 2l+2 resolves each potential conflict between a vertex of A and a vertex of B. Because the b i are in order of increasing partial sum in H 2l+2, the labeling of H 3 resolves all potential conflicts within B. Similarly, since the labeling of H 2l+2 gives the same partial sum to all vertices of A, the labeling of H 3 resolves all potential conflicts within A. We have checked that the labeling is antimagic. 5

6 3 Regular bipartite graphs with even degree Lemma 8. Let n be a positive integer. If n is even, then we can partition {1, 2,..., 3n} into triples such that the sum of each triple is 6n + 3 or 3n. If n is odd, then we can partition {1, 2,..., 3n} into triples such that the sum of each triple is 6n or 3n. Furthermore, each triple consists of one integer from each residue class (mod3). Proof: Suppose n is even. We partition the labels into triples so that the sum of each triple is either 3n or 6n + 3. Consider the triples (3n 3i + 3, 3n 3i + 2, 6i 2) and (3i, 3i 1, 3n 6i + 1) for 1 i n/2. Triples of the first type sum to 6n + 3 and triples of the second type sum to 3n. Suppose n is odd. We partition the labels into triples so that the sum of each triple is either 3n or 6n. Consider the triples (3n 3i+3, 3n 3i+2, 6i 5) for 1 i n/2 and (3i, 3i 1, 3n 6i + 1) for 1 i n/2. Triples of the first type sum to 6n and triples of the second type sum to 3n. Theorem 9. Every regular bipartite graph of even degree at least 8 is antimagic. Proof: We decompose G into two 3-factors and a (2l + 2)-factor; call these G 3, H 3, and H 2l+2, respectively. We label H 2l+2 as in Lemma 5, using all but the 6n smallest labels. This resolves every conflict between a vertex of A and a vertex of B. We partition the labels {3n + 1, 3n + 2,..., 6n} into triples as in Lemma 8. In G 3, at each vertex of A we will use the three labels of some triple. To ensure the sum at each vertex of B is 0(mod 3), we do the following. Partition the 3-factor into three 1-factors; We use 0-labels on the first 1-factor, 1-labels on the second 1-factor, and 2-labels on the third 1-factor. Now consider the partial sums in the union of H 2l+2 and G 3 ; let b i denote the vertices of B in order of increasing partial sum. Label H 3 as in Lemma 6. This resolves every conflict between two vertices in the same part. Hence, the labeling is antimagic. Lemma 10 is very similar to Lemma 8. Lemma 10 serves the same role in the proof of Theorem 11 that Lemma 8 does in the proof of Theorem 9. Lemma 10. Let n be a positive integer. Let H be the set of positive labels less than 4n that are not 0(mod 4), i.e. H = {1, 2, 3, 5, 6,..., 4n 2, 4n 1}. If n is even, then we can partition H into triples such that the sum of each triple is either 4n 2 or 8n + 2. If n is odd, then we can partition H into triples such that the sum of each triple is either 4n 2 or 8n 2. Furthermore, each triple consists of one integer from each nonzero residue class (mod 4). Proof: Suppose n is even. We have triples of the form (8i 3, 4n 4i + 2, 4n 4i + 3), with 1 i n/2, and triples of the form (4n 8i + 1, 4i 2, 4i 1), with 1 i n/2. It is easy to see that triples of the first form sum to 8n + 2 and that triples of the second form sum to 4n 2. It is straightforward to verify that these triples partition H. 6

7 Suppose n is odd. We have triples of the form (8i 7, 4n 4i + 2, 4n 4i + 3), with 1 i n/2, and triples of the form (4n 8i+1, 4i 2, 4i 1), with 1 i n/2. It is easy to see that triples of the first form sum to 8n 2 and that triples of the second form sum to 4n 2. It is straightforward to verify that these triples partition H. Theorem 11. Every 6-regular bipartite graph is antimagic. Proof: Throughout this proof we assume that n is odd. The argument is analagous when n is even, so we omit the details. We decompose G into a 1-factor, a 2-factor, and a 3-factor; call these H 1, H 2, and H 3, respectively. We label H 3 with the labels that are less than 4n and are not 0 (mod 4), so that the partial sum at each vertex of B is 2(mod 4) and the partial sum at each vertex of A is 4n 2 or 8n 2. To do this we partition the labels for H 3 into triples as specified in Lemma 10. At each vertex of A, we use the three labels in some triple. More exactly, we decompose H 3 into three 1-factors; we use 1(mod 4) labels on the first 1-factor, use 2( mod 4) labels on the second 1-factor, and use 3( mod 4) labels on the third 1-factor. This ensures that the partial sum at each vertex of B is 2(mod 4). We label H 2 with the labels 4n + 1 through 6n, so that the partial sum at each vertex of A is 10n + 1 and the sum at each vertex of B is 10n + 1(mod 4). To do this, we partition the labels for H 2 into pairs that sum to 10n + 1. We consider the labels in each pair (mod 4). We have two types of pairs: (1, 2) pairs and (3, 0) pairs. We want to avoid using two labels at a vertex of B that sum to 3(mod 4). We choose the pairs of labels to use on each cycle arbitrarily, except that each cycle must use at least one (1, 2) pair and at least one (3, 0). On each cycle, we first use all the allotted (1, 2) pairs, alternating them as (1, 2), (2, 1), (1, 2), (2, 1),..., then use all the allotted (3, 0) pairs, alternating them as (3, 0), (0, 3), (3, 0), (0, 3),.... As long as we use at least one (1, 2) pair and one (3, 0) pair on each cycle of H 2, we have no problems. This is easy to do, so we are able to avoid vertex-sums in B that are congruent to 3(mod 4). Now we consider partial sums in H 2 H 3. The partial sum at each vertex of A is 14n 1 or 18n 1; since n is odd, the partial sum at each vertex of A is congruent to 1(mod 4). The partial sum at each vertex of B is not congruent to 1(mod 4). The labels we will use on H 1 are all multiples of 4. Hence, regardless of how we label H 1, no vertex in A will conflict with any vertex in B. We call a vertex in A small if it s partial sum in H 2 H 3 is 14n 1; otherwise, we call it big. It is clear that regardless of how we label H 1, no big vertex will conflict with another big vertex; similarly, no small vertex will conflict with a small vertex. Observe that the largest possible sum at a small vertex is 14n 1 + 4n = 18n 1. The smallest possible sum at a big vertex is 18n = 18n + 3. Hence, no small vertex will conflict with a big vertex. Thus, we choose the labels for H 1 to ensure that no two vertices in B conflict. Let b i denote the vertices of B in order of increasing partial sum in H 2 H 3. In H 1, we use label 4i at vertex i. This ensures that vertex-sums in B are distinct. Thus, the labeling is antimagic. The proof for 4-regular graphs is more complicated than for 6-regular graphs. In the 6-regular case, we labeled the 2-factor to ensure there were no conflicts between any 7

9 (2, 2, 3). Thus, we do not swap any labels incident to a light vertex. However, the labels incident to a heavy vertex could become (2, 3, 3) or even (3, 3, 3). In each case though, the vertex remains good. Finally, if any vertex in A is adjacent to two or more bad vertices, then we swap the labels on its incident edges to make each vertex good. Thus, we have at most n/9 bad vertices and each vertex in A is adjacent to at most one bad vertex. Now we assign each edge an actual label (rather than only a residue class) so that the partial sum at each bad vertex is small. We assign the n/9 smallest 1(mod 4) labels to be incident to the bad vertices; the largest is less than 4n/9. Similarly, we assign the n/9 smallest 2(mod 4) labels to be incident to the bad vertices; again the largest is less than 4n/9. Each time we assign a label, we also assign the other labels in its triple. Since each 2( mod 4) label is in a triple with the 3( mod 4) label one greater, the n/9 smallest 3(mod 4) labels are already assigned. So we assign the next n/9 smallest 3( mod 4) labels to be incident to the bad verties; the largest of these labels is less than 8n/9. Finally, we will assign the n/9 smallest 0( mod 4) labels to be incident to the bad vertices. Thus, the largest vertex-sum at a bad vertex is less than 3(4n/9)+8n/9 < 3n. Hence, no bad vertex will conflict with any vertex in A. To ensure that no two bad vertices conflict, we assign the labels to the final 1- factor in order of increasing partial sum at the bad vertices. After we assign all the labels incident to the bad vertices, we assign the remaining labels incident to the good vertices, again in order of increasing partial sum in B. This ensures that no two good vertices conflict. If the partial sum at a vertex of A is 4n 2 we call it small; otherwise we call it big. After we assign the labels on the final 1-factor, the smallest possible vertex-sum at a big vertex is (8n 2) + 4 = 8n + 2; the largest possible sum at a small vertex is (4n 2) + 4n = 8n 2. So no small vertex conflicts with a big vertex. Additionally, all the small vertex-sums are distinct; so are the large vertex-sums. Thus, the labeling is antimagic. 4 Acknowledgements Thanks to Douglas West for help with the writing, and thanks to an anonymous referee, who identified a number of inconsistencies and inaccuracies. References [1] N. Alon, G. Kaplan, A. Lev, Y. Roditty, and R. Yuster, Dense Graphs are Antimagic, J. Graph Theory 47 (2004) [2] N. Hartsfield and G. Ringel, Pearls in Graph Theory: A Comprehensive Introduction. Academic Press, [3] D. Hefetz, Anti-magic graphs via the combinatorial nullstellensatz. J. Graph Theory 50 (2005) [4] D.B. West, Introduction to Graph Theory. Prentice Hall, second edition,

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