Zero-Sum Flow Numbers of Triangular Grids
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1 Zero-Sum Flow Numbers of Triangular Grids Tao-Ming Wang 1,, Shih-Wei Hu 2, and Guang-Hui Zhang 3 1 Department of Applied Mathematics Tunghai University, Taichung, Taiwan, ROC 2 Institute of Information Science Academia Sinica, Taipei, Taiwan, ROC 3 Department of Applied Mathematics National Chung Hsing University, Taichung, Taiwan, ROC Abstract. As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the values of edges are less than k. Note that from algebraic point of view finding such zero-sum flows is the same as finding nowhere zero vectors in the null space of the incidence matrix of the graph. We consider in more details a combinatorial optimization problem, by defining the zero-sum flow number of G as the least integer k for which G admitting a zerosum k-flow. It is well known that grids are extremely useful in all areas of computer science. Previously we studied flow numbers over hexagonal grids and obtained the optimal upper bound. In this paper, with new techniques we give completely zero-sum flow numbers for certain classes of triangular grid graphs, namely, regular triangular grids, triangular belts, fans, and wheels, among other results. Open problems are listed in the last section. 1 Background and Motivation Let G be a directed graph. A nowhere-zero flow on G is an assignment of non-zero integers to each edge such that for every vertex the Kirchhoff current law holds, that is, the sum of the values of incoming edges is equal to the sum of the values of outgoing edges. A nowhere-zero k-flow is a nowhere-zero flow using edge labels with maximum absolute value k 1. Note that for a directed graph, admitting nowhere-zero flows is independent of the choice of the orientation, therefore one may consider such concept over the underlying undirected graph. A celebrated conjecture of Tutte in 1954 says that every bridgeless graph has a nowhere-zero 5-flow. F. Jaeger showed in 1979 that every bridgeless graph has a nowhere-zero 8-flow[6], and P. Seymour proved that every bridgeless graph has a nowhere-zero-6-flow[9] in However the original Tutte s conjecture remains open. The corresponding author whose research is partially supported by the National Science Council of Taiwan under project NSC M J. Chen, J.E. Hopcroft, and J. Wang (Eds.): FAW 2014, LNCS 8497, pp , c Springer International Publishing Switzerland 2014
2 Zero-Sum Flow Numbers of Triangular Grids 265 There is an analogous and more general concept of a nowhere-zero flow that uses bidirected edges instead of directed ones, first systematically developed by Bouchet[4] in Bouchet raised the conjecture that every bidirected graph with a nowhere-zero integer flow has a nowhere-zero 6-flow, which is still unsettled. Recently another related nowhere-zero flow concept has been studied, as a special case of bi-directed one, over the undirected graphs by S. Akbari et al.[2] in Definition 1. For an undirected graph G, a zero-sum flow is an assignment of non-zero integers to the edges such that the sum of the values of all edges incident with each vertex is zero. A zero-sum k-flow is a zero-sum flow whose values are integers with absolute value less than k. Note that from algebraic point of view finding such zero-sum flows is the same as finding nowhere zero vectors in the null space of the incidence matrix of the graph. S. Akbari et al. raised a conjecture for zero-sum flows similar to the Tutte s 5-flow Conjecture for nowhere-zero flows as follows: Conjecture. (Zero-Sum 6-Flow Conjecture) If G is a graph with a zerosum flow, then G admits a zero-sum 6-flow. It was proved in 2010 by Akbari et al. [1] that the above Zero-Sum 6-Flow Conjecture is equivalent to the Bouchet s 6-Flow Conjecture for bidirected graphs. In literatures a more general concept flow number, which is defined as the least integer k for which a graph may admit a k-flow, has been studied for both directed graphs and bidirected graphs. We extend the concept in 2011 to the undirected graphs and call it zero-sum flow numbers, and also considered general constant-sum flows for regular graphs[11]. It is well known that grids are extremely useful in all areas of computer science. One of the main usage, for example, is as the discrete approximation to a continuous domain or surface. Numerous algorithms in computer graphics, numerical analysis, computational geometry, robotics and other fields are based on grid computations. Also it is known that there are three possible types of regular tessellations, which are tilings made up of squares, equilateral triangles, and hexagons. Formally, a lattice grid, or a lattice grid graph is induced by a finite subset of the infinite integer lattice grid Z Z. The vertices of a lattice grid are the lattice points, and the edges connect the points which are at unit distance from each other. The infinite grid Z Z may be viewed as the set of vertices of a regular tiling of the plane with unit squares. There are only two other types of plane tiling with regular polygons. One is with equilateral triangles, which defines an infinite triangular grid in the same way. A triangular grid graph is a graph induced by a finite subset of the infinite triangular grid. The other type of tiling is with regular hexagons which defines an infinite hexagonal grid. Similarly the graph induced by a finite subset of the infinite hexagonal grid is called a hexagonal grid graph. A hexagonal graph is also named a honeycomb graph in literature.
3 266 T.-M. Wang, S.-W. Hu, and G.-H. Zhang In this paper, we calculate zero-sum flow numbers for some classes of triangular grid graphs. In particular we consider the problem for two classes of well known generalized triangular grid graphs, namely fans and wheels. 2 Zero-Sum Flow Numbers In the study of nowhere-zero flows of directed graphs(bidirected graphs) one considers a more general concept, namely, the least number of k for which a graph may admit a k-flow. In 2011 [11] we consider similar concepts for zerosum k-flows: Definition 2. Let G be a undirected graph. The zero-sum flow number F (G) is defined as the least number of k for which G may admit a zero-sum k-flow. F (G) = if no such k exists. Obviously the zero-sum flow numbers can provide with more detailed information regarding zero-sum flows. For example, we may restate the previously mentioned Zero-Sum Conjecture as follow: Suppose a undirected graph G has a zero-sum flow, then F (G) 6. In 2012, we showed some general properties of small flow numbers, so that the calculation of zero-sum flow numbers gets easier. It is well known that a graph admits a nowhere-zero 2-flow if and only if it is Eulerian (every vertex has even degree). We obtain the following for zero-sum flows: Lemma 3 ((T. Wang and S. Hu [12])). A graph G has zero-sum flow number F (G) =2if and only if G is Eulerian with even size (even number of edges) in each component. Tutte obtained in 1949 that a cubic graph has a nowhere-zero 3-flow if and only if it is bipartite. Similarly for undirected graphs we have that: Lemma 4 ((T. Wang and S. Hu [12])). A cubic graph G has zero-sum flow number F (G) =3if and only if G admits a perfect matching. Lemma 5. Let G be a undirected graph and G = H 1 H 2 be an arbitrary union of H 1 and H 2,whereflownumbersF(H 1 )=k 1 and F (H 2 )=k 2.Then F (G) k 1 k 2. Proof. Since F (H 1 )=k 1 and F (H 2 )=k 2, we have two edge labeling functions f 1 : E(H 1 ) {±1,, ±(k 1 1)} and f 2 : E(H 2 ) {±1,, ±(k 2 1)}, which are zero-sum k i -flow for H i, i =1, 2. To make an edge labeling for G, we set f1 (e) =f 1 (e) ife E(H 1 ), and f1 (e) = 0, otherwise. Also f2 (e) =f 2 (e) if e E(H 2 ), and f2 (e) = 0 otherwise, for all e E(G). Now, let f = f 1 + k 1f2 or k 2 f1 + f 2, then it can be seen that the function f forms a zero-sum k 1k 2 -flow for G. Then we obtain the following corollaries, which are extremely useful tools for calculating flow numbers in general:
4 Zero-Sum Flow Numbers of Triangular Grids 267 Corollary 6. Let G be a undirected graph and G = H 1 H 2 be an arbitrary union of H 1 and H 2,whereflownumbersF (H 1 )=F (H 2 )=2.ThenF (G) 4. Corollary 7. Let G be a undirected graph and G = H 1 H 2 be an arbitrary union of H 1 and H 2,whereflownumbersF (H 1 )=2and F (H 2 )=3.Then F (G) 6. We calculated in 2013 [13] for the zero-sum flow numbers of hexagonal grids. Note that Akbari. et al. showed that in [2] if Zero-Sum 6-Flow Conjecture is true for (2, 3)-graphs (in which every vertex is of degree 2 or 3), then it is true for any graph. Therefore the study can be reduced to (2, 3)-graphs. It is clear non-trivial hexagonal grid graphs are a special class of (2, 3)-graphs. In particular we found the optimal upper bound and also provided with infinitely many examples of hexagonal grid graphs with flow number 3 and 4 respectively. 3 Flow Numbers for Regular Triangular Grids Contrast to the case of hexagonal grids, the upper bound for the flow numbers of an arbitrary triangular grid is somewhat difficult to find. In this section we initiate the study and calculate the exact value of zero-sum flow numbers for the class of regular triangular grids as defined in below. Definition 8. Let G be a triangular grid stacked by n layers of triangles with 1, 2,,n triangles in each layer. We call G a regular triangular grid and denote it by T n. For example see Figure 1 for T 5. Fig. 1. The regular triangular grid T 5 Theorem 9. The flow numbers of T n are as follows:, n =1. 2, n 3, 4(mod 4). F (T n )= 3, n 1, 2(mod 4). 4, n =2.
5 268 T.-M. Wang, S.-W. Hu, and G.-H. Zhang Proof. Note that T 1 has no zero-sum flow, thus F (T 1 )=. Then we consider the following for the remaining cases. Case 1. n 3, 4(mod 4). Note that in general there is no any odd degree vertex in T n (the only possible degrees are 2, 4, and 6), and we see E(T n ) = 3n(n+1) 2.Sowhenn 3, 4(mod 4), E(T n ) is even. Then by Lemma 3, F (T n )=2inthiscase. Case 2. n 1, 2(mod 4). As seen in Case 1, T n is an even graph. But now when n 1, 2(mod 4), E(T n ) is odd. Therefore by Lemma 3, we see F (T n ) 2 in this case. Assume first n 1(mod 4), i.e. n =4s +1 forsome s, we find two special even subgraphs that are both of even size. We treat T 4s+1 as the union of T 4s and H as in Figure 2, where the intersection of the edge sets for T 4s and H is the triangle v 1 v 2 v 3. We see the size of H equals to 3n +3 = 12s + 6, which is even, thus F (H) =2 by Lemma 3. On the other hand, by Case 1 we see F (T 4s )=2.Nowwejust need to focus on the labels of v 1 v 2,v 2 v 3,v 3 v 1.Weusethelabels1and-1asin Figure 2 over two Euler paths on T 4s and H respectively, to make the labels of v 1 v 2,v 2 v 3,v 3 v 1 in T 4s coincide with that of v 1 v 2,v 2 v 3,v 3 v 1 in H. -1 Fig. 2. Two (1, -1)-labeled Euler paths over T 4s and H Then we see that, by extending these labels to the whole T 4s, together with H, which give a zero-sum 3-flow for T 4s+1, with labels over v 1 v 2,v 2 v 3,v 3 v 1 are 2, 2, -2, and 1, -1 everywhere else. Note that similarly T 4s+2 equals to T 4s+1 H,whereH is the edge disjoint union of 4s + 2 copies of triangles. Now T 4s+1 part just follow the labels we built previously. Note that the H part is an even graph with even size, whose flow number is 2. Therefore we give a zero-sum 3-flow for T 4s+2. Case 3. n =2. T 2 is union of two even graphs with even size as in Figure 3. Then by Corollary 6, one see F (T 2 ) 4. First by Lemma 3 we have F (T 2 ) 2. We want to show further that F (T 2 ) 3. Suppose a, b, c give a zero-sum 3-flow over T 2 as in Figure 4.
6 Zero-Sum Flow Numbers of Triangular Grids 269 Fig. 3. T 2 as union of two subgraphs Fig. 4. Labels over T 2 Since it is a zero-sum 3-flow with a, b, c are distinct, there are only 4 possibilities for (a, b, c): (1, 1, 2), ( 1, 1, 2), (1, 2, 2), ( 1, 2, 2). But all these four cases make c b = 3, a contradiction. Hence F (T 2 )=4. 4 Flow Numbers for Triangular Belts Definition 10. A triangular belt is exactly the square of paths P 2 n, n 3, in which we connect distance two vertices in a path P n,asinfigure5. Fig. 5. A triangular belt Note that P3 2 = C 3 and P4 2 = F 3, and both admit no any zero-sum flow (The fan graph F 3 is defined and seen in Figure 16 next section). We start from P5 2. A zero-sum 3-flow for P5 2 is obtained from adding the corresponding edge labels together with two basic labeled parallelograms as in Figure 6. Theorem 11. The flow number of the graph square of paths F (Pn)=3when 2 n 5, andf (Pn 2 )= for n =3, 4. Proof. We proceed with induction on n 5. Since there are odd degree vertices, F (P 2 5 ) = 3. Suppose that f is a zero-sum 3-flow for P 2 n 1, n 6, as in Figure 7, which is obtained by using two basic labeled parallelograms as in Figure 6.
7 270 T.-M. Wang, S.-W. Hu, and G.-H. Zhang Fig. 6. Azero-sum3-flowforP 2 5 Fig. 7. Azero-sum3-flowforP 2 n 1 Then we consider Pn 2 as the union of a C 4 with the Pn 1 2, with intersection over one edge, as shown in Figure 8. Note that Pn 2 = Pn 1 2 C 4 admits a zerosum 3-flow and it has odd degree vertices, hence F (Pn)=3. 2 Fig. 8. P 2 n 1 C 4 5 Flow Numbers for Fans and Wheels In order to obtain the index sets of fans and wheels, we first describe a subdivision method which is commonly used here for the construction of zero-sum flows, in particular fans and wheels. Triple Subdivision Method Let G beagraphadmittingazero-sumflowf. Using the triple subdivision method we may obtain a new graph G with larger order, and a new zero-sum flow f of G, based upon G and f. We proceed by choosing in G avertexv and edges e 1,e 2 with f(e 1 )=f(e 2 )=x, which are not incident with v. Then subdivide these two edges by inserting new vertices of degree 3, join them to v respectively. See Figure 9. Now then we may construct a new f on G by keeping
8 Zero-Sum Flow Numbers of Triangular Grids 271 Fig. 9. Triple Subdivision Method the labels on G unchanged, and labeling x, x, 2x on three newly inserted edges respectively. Note that x + x 2x = 0, then the new labeling f on G is still zero-sum flow, and f (E(G )) = f(e(g)) { 2x}. Note that fans and wheels are graphs made from joining one point to a path and a cycle respectively as in Figure 10. Also both graphs consist of triangles, and can be seen as special triangular grids. In below, we use the above triple subdivision method by new edge insertion and induction to justify the results for the flow numbers of fans and wheels. Fig. 10. Fans and Wheels Theorem 12. The flow numbers of fans F n and wheels W n are as follows:, n =1, 2, 3. F (F n )= 3, n =3k +1, k 1. 4, otherwise. 3, n =3k, k 1. F (W n )= 5, n =5. 4, otherwise. To prove the above result, we start with the following lemma: Lemma 13. W n does not admit any zero-sum 3-flow for 3 n, andf n does not admit any zero-sum 3-flow for 3 n 1. Proof. Assume F n or W n admits a zero-sum 3-flow, then only ±1, ±2 canbe used for the edge labels. One may see that along the outer cycle of W n or outer
9 272 T.-M. Wang, S.-W. Hu, and G.-H. Zhang path of F n, the 1-labeled edge can be incident to 1-labeled or (-2)-labeled edge only. Similarly (-2)-labeled edge can be incident to 1-labeled or (-2)-labeled edge only. By symmetry, (-1)-labeled and 2-labeled are only possible incident edges. Without loss of generality, may assume there are x 1 s and y (-2) s. Then we have that x 2y =0andx + y = n for W n,orhavethatx 2y =0and x + y = n 1forF n. Therefore if W n admits a zero-sum 3-flow, then 3 n, and if F n admits a zero-sum 3-flow, then 3 n 1. By the technique of triple subdivision, we have the following: Lemma 14. F 3n+1 and W 3n admit zero-sum 3-flows, for n 1, respectively. Proof. By induction from F 3n+1 to F 3n+4 for n 1, and from W 3n to W 3n+3 for n 1 respectively. See the Figure 11 and Figure 12 below, which are triple subdivision (edge insertion) from F 4 to F 7 and from W 3 to W 6 respectively. Fig. 11. A zero-sum 3-flow from F 4 to F 7 Fig. 12. A zero-sum 3-flow from W 3 to W 6 Similarly we may obtain by the same triple subdivision that F 3n+2, F 3n+3 and W 3n+1, W 3n+2 admit zero-sum 4-flows, for n 1, respectively, with the only exception W 5. See lemmas in below. Lemma 15. F 3n+2, F 3n+3 and W 3n+1, W 3n+2 admit zero-sum 4-flows, for n 1, respectively, with the only exception W 5. Proof. See Figure 13 and Figure 14, and one may have zero-sum 4-flows by triple subdivision and induction method. The following is the only exceptional case W 5 :
10 Zero-Sum Flow Numbers of Triangular Grids 273 Fig. 13. Zero-sum 4-flows from F 5 and F 6 respectively Fig. 14. Zero-sum 4-flows from W 4 and W 8 respectively Lemma 16. W 5 does not admit any zero-sum 4-flow. Proof. Suppose it does admit a zero-sum 4-flow. See the Figure 15, where we assume the edges of outside cycle are O i for 1 i 5, and the edges of inner spokes are O i for 1 i 5. Note that only ±1, ±2, ±3 can be used to give the 4- flow, and also by adding zero-sums over every vertex one has Σ 5 i=1 O i = 0. Hence the sum of three consecutively adjacent edge labels must obey 1 O i + O i+1 + O i+2 3 for any i,modulo5,sinceo i +O i+1 +O i+2 = (O i+3 +O i+4 )= I i+3. Fig. 15. W 5 admits a zero-sum 5-flow Without loss of generality we need only consider three cases O 1 =1,O 1 =2, or O 1 =3.IfO 1 = 1, then the only possible labels on two incident edges O 2 and O 5 are 1, 2, 2, 3. So the possible pairs of (O 2,O 5 ) by symmetry are (1, 1), (1, 2), (1, 2), (1, 3), (2, 2), (2, 2), (2, 3), ( 2, 2), ( 2, 3), ( 3, 3). We may rule out cases (1, 2), (1, 2), (2, 2), (2, 3), ( 2, 3), ( 3, 3) by using the condition 1 O i + O i+1 + O i+2 3. On the other hand, we exhausted the discussion over all remaining cases. Say in case the pair (O 2,O 5 )=(1, 1), then
11 274 T.-M. Wang, S.-W. Hu, and G.-H. Zhang the pair (O 3,O 4 ) has to be either ( 1, 2) or ( 2, 1). However this would endupwithsomei i equals to 0, a contradiction. Similarly one would reach the contradiction for the rest of cases (O 2,O 5 )=(1, 3), (2, 2), or ( 2, 2), and also for the cases when O 1 =2andO 1 = 3. Therefore we are done with all possibilities, and obtain that W 5 does not admit any zero-sum 4-flow. It is clear to see that F 1 has degree 1 vertex and F 2 = C3,soF (F 1 )andf (F 2 ) are both infinity. F 3 case are as follows: Fig. 16. F 3 Suppose F 3 has flow number r, asinfigure16.thenx + y + z should equal to r. On the other hand, r x + z + r y equals to r as well. Thus r + z = x + y, but r = x + y + z. Therefore z = 0, a contradiction. This shows F (F 3 )=. Therefore with above Lemmas and the existence of labeling of zero-sum 4- flows and zero-sum 5-flows, we finish calculating the flow numbers of fans and wheels for Theorem Concluding Remark One may further study various classes of triangular grid graphs. Contrast to the case of hexagonal grids, the upper bound for the flow numbers of an arbitrary triangular grid is somewhat difficult to find. We list two problems as follows: 1. Give the optimal upper bound for zero-sum flow numbers of general triangular grid graphs and classify them; 2. Give the optimal upper bound for zero-sum flow numbers of various classes of planar graphs and classify them. References [1] Akbari, S., Daemi, A., Hatami, O., Javanmard, A., Mehrabian, A.: Zero-Sum Flows in Regular Graphs. Graphs and Combinatorics 26, (2010) [2] Akbari, S., Ghareghani, N., Khosrovshahi, G.B., Mahmoody, A.: On zero-sum 6-flows of graphs. Linear Algebra Appl. 430, (2009) [3] Akbari, S., et al.: A note on zero-sum 5-flows in regular graphs. The Electronic Journal of Combinatorics 19(2), P7 (2012) [4] Bouchet, A.: Nowhere-zero integral flows on a bidirected graph. J. Combin. Theory Ser. B 34, (1983)
12 Zero-Sum Flow Numbers of Triangular Grids 275 [5] Gallai, T.: On factorisation of grahs. Acta Math. Acad. Sci. Hung. 1, (1950) [6] Jaeger, F.: Flows and generalized coloring theorems in graphs. J. Combin. Theory Ser. B 26(2), (1979) [7] Kano, M.: Factors of regular graph. J. Combin. Theory Ser. B 41, (1986) [8] Petersen, J.: Die Theorie der regularen graphs. Acta Mathematica (15), (1891) [9] Seymour, P.D.: Nowhere-zero 6-flows. J. Combin. Theory Ser. B 30(2), (1981) [10] Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, (1954) [11] Wang, T.-M., Hu, S.-W.: Constant Sum Flows in Regular Graphs. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM LNCS, vol. 6681, pp Springer, Heidelberg (2011) [12] Wang, T.-M., Hu, S.-W.: Zero-Sum Flow Numbers of Regular Graphs. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds.) AAIM 2012 and FAW LNCS, vol. 7285, pp Springer, Heidelberg (2012) [13] Wang, T.-M., Zhang, G.-H.: Zero-Sum Flow Numbers of Hexagonal Grids. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM LNCS, vol. 7924, pp Springer, Heidelberg (2013)
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