On bipartite graphs of defect at most 4
aa r X i v : . [ m a t h . C O ] J un On bipartite graphs of defect at most 4
Ramiro Feria-Pur´on ∗ School of Electrical Engineering and Computer ScienceThe University of Newcastle, Australia
Guillermo Pineda-Villavicencio † Centre for Informatics and Applied OptimisationUniversity of Ballarat, Australia
October 9, 2018
Abstract
We consider the bipartite version of the degree/diameter problem , namely, given natural numbers∆ ≥ D ≥
2, find the maximum number N b (∆ , D ) of vertices in a bipartite graph of maximumdegree ∆ and diameter D . In this context, the Moore bipartite bound M b (∆ , D ) represents an upperbound for N b (∆ , D ).Bipartite graphs of maximum degree ∆, diameter D and order M b (∆ , D ) – called Moore bipartitegraphs – turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs ofmaximum degree ∆ ≥
2, diameter D ≥ b (∆ , D ) − ǫ with small ǫ >
0; that is, bipartite(∆ , D, − ǫ )-graphs. The parameter ǫ is called the defect .This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs.Bipartite graphs of defect 2 have been studied in the past; if ∆ ≥ D ≥
3, they may only existfor D = 3. However, when ǫ > , D, − ǫ )-graphs represent a wide unexplored area.The main results of the paper include several necessary conditions for the existence of bipartite(∆ , d, − , D, − ǫ )-graphs with D ≥ ≤ ǫ ≤ , D, − ǫ )-graphs with ∆ ≥
2, 5 ≤ D ≤
187 ( D = 6) and0 ≤ ǫ ≤
4; and a non-existence proof of all bipartite (∆ , D, − ≥ D ≥ ≥ D ≥
5, andcomment on some implications of our results for the upper bounds of N b (∆ , D ). ∗ [email protected] † Corresponding author: [email protected] eywords: Moore bipartite bound; Moore bipartite graph; Degree/diameter problem for bipartite graphs;defect; repeat.
AMS Subject Classification:
Due to the diverse features and applications of interconnection networks, it is possible to find manyinterpretations about network “optimality” in the literature. Here we are concerned with the following;see [8, pp. 18], [10, pp. 168], and [16, pp. 91 ].
An optimal network contains the maximum possible number of nodes, given a limit on thenumber of connections attached to a node and a limit on the distance between any two nodesof the network.
This interpretation has attracted network designers and the research community in general due to itsimplications in the design of large interconnection networks. In graph-theoretical terms, this interpretationleads to the degree/diameter problem (the problem of finding the largest possible number of vertices ina graph with given maximum degree and diameter). If the graphs in question are subject to furtherrestrictions such as being bipartite, planarity and/or transitivity, we can state the degree/diameter problemfor the classes of graphs under consideration.In this paper we will consider only bipartite graphs, and in this case, the degree/diameter problem canbe stated as follows.
Degree/diameter problem for bipartite graphs : Given natural numbers ∆ ≥ D ≥
2, find thelargest possible number N b (∆ , D ) of vertices in a bipartite graph of maximum degree ∆ and diameter D .Note that N b (∆ , D ) is well defined for ∆ ≥ D ≥
2. An upper bound for N b (∆ , D ) is given by the Moore bipartite bound M b (∆ , D ), defined below:M b (∆ , D ) = 2 (cid:0) −
1) + · · · + (∆ − D − (cid:1) . Bipartite graphs of degree ∆, diameter D and order M b (∆ , D ) are called Moore bipartite graphs . Moorebipartite graphs are rare; for ∆ = 2 they are the cycles of length 2 D , while for ∆ ≥ D and order M b (∆ , D ) − ǫ for ǫ > , D, − ǫ )-graphs, where the parameter ǫ is called the defect . For notational convenience,we consider Moore bipartite graphs as having defect ǫ = 0.Only a few values of N b (∆ , D ) are known at present. With the exception of N b (3 ,
5) = M b (3 , − b (∆ , D ) are those for which there is a Moore bipartite graph.The paper [14] combined with [5, 6] almost settled the case of bipartite graphs of defect 2; if ∆ ≥ D ≥
3, then such graphs may only exist for D = 3 and certain values of ∆. Bipartite (∆ , D, − ǫ )-graphswith ǫ > , D, − ≥ D ≥
3. By using combinato-rial approaches we obtain several important results about bipartite graphs of defect 4, including severalnecessary conditions for the existence of bipartite (∆ , d, − , D, − ǫ )-graphs with D ≥ ≤ ǫ ≤
4; the complete catalogue of bipartite (∆ , D, − ǫ )-graphs with∆ ≥
2, 5 ≤ D ≤
187 ( D = 6) and 0 ≤ ǫ ≤
4; and a non-existence proof of all bipartite (∆ , D, − ≥ D ≥
5. Finally, we conjecture that there are no bipartite graphs of defect 4 for ∆ ≥ D ≥ , D, − ≥ D = 3 , , D, − ≥ D ≥
3. At the time of writing the paper we do not foresee a conclusive way to take onthe diameters 3 and 4. To deal with such graphs it would be necessary to either find different ideas orcomplement some of the ones presented here. Section 6.1 contains further comments on such diameters.
The terminology and notation used in this paper is standard and consistent with that used in [7], so onlythose concepts that can vary from texts to texts will be defined.All graphs considered are simple. The vertex set of a graph Γ is denoted by V (Γ), and its edge setby E (Γ). The difference between the graphs Γ and Γ ′ , denoted by Γ − Γ ′ , is the graph with vertex set V (Γ) − V (Γ ′ ) and edge set formed by all the edges with both endvertices in V (Γ) − V (Γ ′ ).The set of neighbours of a vertex x in Γ is denoted by N ( x ). For an edge e = { x, y } we write e = xy ,or alternatively x ∼ y . The set of edges in a graph Γ joining a vertex x in X ⊆ V (Γ) to a vertex y in Y ⊆ V (Γ) is denoted by E ( X, Y ); for simplicity, we write E ( x, Y ) rather than E ( { x } , Y ).A path of length k is called a k -path , and cycle of length k is called a k -cycle . A path from a vertex x to a vertex y is denoted by x − y . Whenever we refer to paths we mean shortest paths. We will use thefollowing notation for subpaths of a path P = x x . . . x k : x i P x j = x i . . . x j , where 0 ≤ i ≤ j ≤ k . Thedistance between a vertex x and a vertex y is denoted by d ( x, y ).3he union of three independent paths of length D with common endvertices is denoted by Θ D . In agraph Γ a vertex of degree at least 3 is called a branch vertex of Γ. (∆ , D, − ǫ ) -graphs with ∆ ≥ , D ≥ and ≤ ǫ ≤ For ∆ = 2 the Moore bipartite graphs are the cycles on 2 D vertices, while for D = 2 and each ∆ ≥ D = 3 , , − −
1. The question of whether Moore bipartite graphs of diameter 3, 4 or 6 exist for othervalues of ∆ remains open, and represents one of the most famous problems in combinatorics. For othervalues of D ≥ ≥ D = 2 bipartite (∆ , D, − ǫ )-graphs with ǫ ≥ D ≥ , D, − ǫ )-graph with ǫ ≥
1: the path of length D , which hasdefect ǫ = D −
1. For a given ∆ ≥ − , , − ǫ )-graphs with ǫ ≥
1; theyare the complete bipartite graphs with partite sets of size ∆ and ∆ − ǫ , where 1 ≤ ǫ ≤ ∆ −
1. Therefore,from now on we assume ∆ ≥ D ≥ , D, − ǫ )-graphs, which were obtainedin [5]. Proposition 3.1 ([5])
For ǫ < −
1) + (∆ − + . . . + (∆ − D − , ∆ ≥ and D ≥ , a bipartite (∆ , D, − ǫ ) -graph is regular. Proposition 3.2 ([5])
For ǫ < (cid:0) (∆ −
1) + (∆ − + . . . + (∆ − D − (cid:1) , ∆ ≥ and odd D ≥ , abipartite (∆ , D, − ǫ ) -graph is regular. By Propositions 3.1 and 3.2, bipartite (∆ , D, − ǫ )-graphs with ∆ ≥ D ≥ ǫ ≤ ≥ D ≥ ǫ = 1 ,
3. In the same way, bipartite(∆ , D, − ≥ D ≥ , , − ≥ ≥ D ≥
3, the only known bipartite (∆ , D, − ≥ D ≥
4; see [5, 6, 14].Bipartite (3 , , − geng from the package nauty written by McKay [13]. The unique bipartite (3 , , − a ) ( b ) Figure 1: ( a ) the unique bipartite (3 , , − b ) the unique bipartite (4 , , − ( a ) ( b ) ( c ) ( d ) Figure 2: All the bipartite (3 , , − , , − b )( a ) Figure 4: All the bipartite (4 , , − , , − , , − genreg . An alternative description of the graph in Fig. 4 ( b ) wascommunicated to the second author by Charles Delorme: take Z / Z as the vertex set of the graph, andfor each even x , add the edges { x, x + 1 } , { x, x − } , { x, x + 7 } and { x, x + 11 } .The only known bipartite (5 , , − Z / Z as its vertex set, and for each even x , add the edges { x, x − } , { x, x + 1 } , { x, x + 5 } , { x, x + 13 } and { x, x + 23 } . From now on we use the symbol d rather than ∆ to denote the degree of a regular graph, as it is customary.Recall that, unless d = 3 and D = 3, a bipartite ( d, D, − ≥ D ≥ d, D, − ≥ D ≥
3, we are actuallyreferring to any bipartite ( d, D, − ≥ D ≥ c ) and ( d ).In a bipartite ( d, D, − D − short cycle . Proposition 4.1
The girth of a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ is D − .Furthermore, any vertex x of Γ lies on the short cycles specified below and no other short cycle, and wehave the following cases: x is contained in exactly three (2 D − -cycles . Then ( i ) x is a branch vertex of one Θ D − , or x is contained in two (2 D − -cycles . Then ( ii ) x lies on exactly two (2 D − -cycles , whose intersection is a ℓ -path with ℓ ∈ { , . . . , D − } . Each case is considered as a type. For instance, a vertex satisfying case ( i ) is called a vertex of Type ( i ).Note that, if x is of Type ( ii ) and ℓ = D −
1, the two short cycles containing x constitute a Θ D − . Proof.
Let xy be an edge of Γ. Let us use the standard decomposition for a bipartite graph of even girthwith respect to the edge xy [3]. For 0 ≤ i ≤ D −
1, the sets X i and Y i are defined as follows: X i = { z ∈ V (Γ) | d ( x, z ) = i, d ( y, z ) = i + 1 } Y i = { z ∈ V (Γ) | d ( y, z ) = i, d ( x, z ) = i + 1 } .7he decomposition of Γ into the sets X i and Y i is called the standard decomposition for a graph of evengirth with respect to the edge xy . Since Γ is bipartite, its girth is even and X i ∩ Y j = ∅ for 0 ≤ i, j ≤ D − Claim 1 g(Γ) = 2 D − Proof of Claim 1.
Since the assertion is trivial for D = 3, we suppose that g(Γ) ≤ D − D ≥ xy lies on a cycle of length g(Γ). Then, | X i | = | Y i | = ( d − i for 1 ≤ i ≤ g(Γ)2 − | X D − | ≤ (cid:0) ( d − D − − (cid:1) ( d −
1) + d − d − D − − | Y D − | ≤ (cid:0) ( d − D − − (cid:1) ( d −
1) + d − d − D − − | X D − | ≤ (cid:0) ( d − D − − (cid:1) ( d − | Y D − | ≤ (cid:0) ( d − D − − (cid:1) ( d − . Therefore, | V (Γ) | = D − X i =0 | X i | + D − X i =0 | Y i | ≤ (cid:0) d −
1) + ( d − + · · · + ( d − D − (cid:1) ++2( d − D − − d − D − − d −
1) == 2 (cid:0) d −
1) + ( d − + · · · + ( d − D − (cid:1) − d − − b ( d, D ) − d, which is a contradiction. Hence, g(Γ) ≥ D −
2. If g(Γ) = 2 D then the order of Γ would be at leastM b ( d, D ) [2]. Thus, g(Γ) = 2 D − ✷ We now proceed to prove the second part of the proposition.For a given vertex x , we use again the standard decomposition for a bipartite graph with respect toan edge xy in Γ. Suppose that there are at least three edges joining vertices at X D − to vertices at Y D − ;that is, | E ( X D − , Y D − ) | ≥ | X D − | ≤ ( d − D − − | Y D − | ≤ ( d − D − − | V (Γ) | = D − X i =0 | X i | + D − X i =0 | Y i | ≤ (cid:0) d −
1) + ( d − + · · · + ( d − D − (cid:1) ++ 2 (cid:0) ( d − D − − (cid:1) == 2 (cid:0) d −
1) + ( d − + · · · + ( d − D − (cid:1) − M bd,D − , ≤ | E ( X D − , Y D − ) | ≤ | E ( X D − , Y D − ) | = 2. If the two edges are both incident to a common vertex of Y D − then x is of Type ( i ), otherwise x is of Type ( ii ).If instead | E ( X D − , Y D − ) | = 1 then | E ( X D − , X D − ) | = | E ( Y D − , Y D − ) | = ( d − D − −
1. Since | X D − | = | Y D − | = ( d − D − −
2, there is a vertex u ∈ X D − such that | E ( u, X D − ) | = 2. Therefore, itfollows ( ii ).Finally, if | E ( X D − , Y D − ) | = 0 then both types may occur. Indeed, if there is a vertex u ∈ X D − suchthat | E ( u, X D − ) | = 3 then x is of Type ( i ) (this case can only occur if d ≥ u, v ∈ X D − such that | E ( u, X D − ) | = | E ( v, X D − ) | = 2, in which case x is of Type ( ii ). Thiscompletes the proof of the proposition. ✷ We continue with the following observation, which will be implicitly used throughout the paper:
Observation 4.1
Let
Γ = ( V ∪ V , E ) (the sets V and V are called partite sets ) be any bipartite graphof even ( odd ) finite diameter D . The distance between a vertex u ∈ V and any vertex v ∈ V ( w ∈ V ) isat most D − . In virtue of Proposition 4.1, we define the following concepts:If two short cycles C and C are non-disjoint we say that C and C are neighbours .For a vertex x lying on a short cycle C , we denote by rep C ( x ) the vertex x ′ in C such that d ( x, x ′ ) = D −
1. We say x ′ is the repeat of x in C and vice versa, or simply that x and x ′ are repeats in C .Alternatively, and more generally, we say that x ′ is a repeat of x with multiplicity m x ( x ′ ) (1 ≤ m x ( x ′ ) ≤ x ( x ′ ) + 1 different paths of length D − x to x ′ . Proposition 4.1 tells us thata vertex in Γ may have a repeat of multiplicity 2. Accordingly, we denote by Rep ( x ) the multiset of therepeats of a vertex x in Γ.The concept of repeat can be easily extended to paths. For a path P = x − y of length at most D − C , we denote by rep C ( P ) the path P ′ ⊂ C defined as rep C ( x ) − rep C ( y ). Wesay that P ′ is the repeat of P in C and vice versa, or simply that P and P ′ are repeats in C .Often our arguments revolve around the identification of the elements in the set S x of short cyclescontaining a given vertex x ; we call this process saturating the vertex x . A vertex x is called saturated if the elements in S x have been completely identified. The following lemma will help us in this cycleidentification process. Lemma 4.1 (Saturating Lemma)
Let C be a (2 D − -cycle in a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ , and α, α ′ two vertices in C such that α ′ = rep C ( α ) . Let γ be a neighbour of α not ontained in C , and µ , µ , . . . , µ d − the neighbours of α ′ not contained in C . Suppose there is no shortcycle in Γ containing the edge α ∼ γ and intersecting C at a path of length greater than D − .Then, in Γ there exist a vertex µ ∈ { µ , µ , . . . , µ d − } and a short cycle C such that γ and µ are repeatsin C , and C ∩ C = ∅ . Proof.
Let α ′ , α ′ be the neighbours of α ′ contained in C .For 1 ≤ i ≤ d −
2, consider the path P i = γ − µ i . As g(Γ) = 2 D − P i cannot go through α ′ or α ′ . If P i went through certain µ j ( j = i ), then a cycle γP i µ j α ′ C αγ would either have length smaller than2 D − C at a ( D − P i must go through one of theneighbours of µ i other than α ′ , and must be a ( D − a ). In addition, V ( P i ∩ C ) = ∅ . αα ′ γµ µ µ d − C α ′ α ′ ( a ) ( b ) . . . ρ αα ′ C α ′ α ′ P P P d − P k Q γµ k ρ C Figure 6: Auxiliary figure for Lemma 4.1Let ρ be one of the neighbours of γ other than α , not contained in any of the paths P i (there is atleast one of such vertices). Consider a path Q = ρ − α ′ . If Q went through α ′ , then the closed walk ρQα ′ C αγρ would either contain a cycle of length smaller than 2 D − C at a ( D − Q must go through a certain µ k (1 ≤ k ≤ d −
2) and V ( Q ∩ C ) = { α ′ } (Fig. 6 ( b )). Note that Q must be a ( D − V ( Q ∩ P k ) = { µ k } ; otherwise there would bea cycle in Γ of length smaller than 2 D − C = γρQµ k P k γ such that γ and µ k are repeats in C , and C ∩ C = ∅ . By setting µ = µ k the lemma follows. ✷ Corollary 4.1
Let α, γ be vertices in a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ suchthat γ ∈ N ( α ) . Then, for every α ′ ∈ Rep ( α ) it follows that N ( α ′ ) contains a repeat of γ . Proof.
Let C be a short cycle containing α and α ′ . If the vertex γ is contained in C or the edge αγ belongsto a short cycle in Γ intersecting C at a path of length D − D −
1, then the corollary trivially follows.10f we instead assume that γ
6∈ C and there is no short cycle in Γ containing the edge αγ and intersecting C at a path of length greater than D −
3, then the corollary follows from the Saturating Lemma. ✷ In this section we extend the concept of repeat to short cycles; see the Repeat Cycle Lemma.
Lemma 4.2 (Repeat Cycle Lemma)
Let C be a short cycle in a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ , { C , C , . . . , C k } the set of neighbours of C , and I i = C i ∩ C for ≤ i ≤ k . Supposeat least one I j , for j ∈ { , . . . , k } , is a path of length smaller than D − . Then there is an additionalshort cycle C ′ in Γ intersecting C i at I ′ i = rep C i ( I i ) , where ≤ i ≤ k . Proof.
Observe that, according to our premises and Proposition 4.1, k ≥ I i ∩ I j = ∅ for 1 ≤ i < j ≤ k .We assume the denotation of the neighbours C , C , . . . , C k of C and the corresponding intersection paths I = x − y , I = x − y , . . . , I k = x k − y k is such that C = x I y x I y . . . x k I k y k x . For 1 ≤ i ≤ k , wealso denote the endvertices of I ′ i by x ′ i and y ′ i , where x ′ i = rep C i ( x i ) and y ′ i = rep C i ( y i ) (see Fig. 7 ( a )). x y y ′ x x ′ CC C x x k y y y k y ′ y ′ y ′ k x ′ x ′ x ′ k C C k ... ... I k I I I I ′ I ′ I ′ I ′ k x y y ′ x x ′ CC C x x k y y y k y ′ y ′ y ′ k x ′ x ′ x ′ k C C k I k I I I I ′ I ′ I ′ I ′ k ... ... ( a ) ( b ) Figure 7: Auxiliary figure for Lemma 4.2For 1 ≤ i ≤ k , consider the cycles C i and C ( i mod k )+1 .First suppose that I i is a path of length smaller than D −
2. Since y i is saturated, there cannot be ashort cycle in Γ, other than C , containing the edge y i ∼ x ( i mod k )+1 . Since I i is a path of length smallerthan D −
2, we apply the Saturating Lemma (mapping C i to C , y i to α , y ′ i to α ′ and x ( i mod k )+1 to γ ) andobtain an additional short cycle C in Γ such that x ( i mod k )+1 is a repeat in C of a neighbour v C i of y ′ i ,and C ∩ C i = ∅ . Since x ( i mod k )+1 is saturated, we have that necessarily C = C ( i mod k )+1 , which in turnimplies v = x ′ ( i mod k )+1 . In other words, it follows that y ′ i ∼ x ′ ( i mod k )+1 ∈ E (Γ).11f instead I i is a ( D − I ( i mod k )+1 must be a path of length smaller than D −
2. Therefore,we can apply the above reasoning and deduce that x ′ ( i mod k )+1 ∼ y ′ i ∈ E (Γ).This way we obtain a subgraph Υ = S ki =1 (cid:0) I ′ i ∪ y ′ i ∼ x ′ ( i mod k )+1 (cid:1) = x ′ I ′ y ′ x ′ I ′ y ′ . . . x ′ k I ′ k y ′ k x ′ intersecting C i at I ′ i for 1 ≤ i ≤ k (see Fig. 7 ( b ), where part of the subgraph Υ is highlighted in bold).We next show that Υ must be indeed a cycle. Claim 1.
Υ is a (2 D − Proof of Claim 1.
If the paths I ′ i are pairwise disjoint then Υ is obviously a (2 D − I ′ i are not pairwise disjoint; then | V (Υ) | < D − z ∈ C ℓ be an arbitrary leaf in Υ. If the repeat path I ′ ℓ = x ′ ℓ − y ′ ℓ had length greater than 0, then z would have at least two neighbours in Υ. Therefore, I ℓ = C ∩ C ℓ contains exactly one vertex, and thus, x ℓ = y ℓ and z = x ′ ℓ = y ′ ℓ .Recall we do addition modulo k on the subscripts of the vertices and the superscripts of the cycles.Since x ′ ℓ ∼ y ′ ℓ − and x ′ ℓ ∼ x ′ ℓ +1 are edges in Υ, it holds that y ′ ℓ − and x ′ ℓ +1 denote the same vertex.Let u ′ ℓ − , v ′ ℓ − be the neighbours of y ′ ℓ − in C ℓ − ; u ′ ℓ +1 , v ′ ℓ +1 the neighbours of x ′ ℓ +1 in C ℓ +1 ; and u ℓ , v ℓ theneighbours of x ℓ in C ℓ . We have that V ( C ℓ − ∩ C ℓ +1 ) = { y ′ ℓ − } , otherwise there would be a third shortcycle in Γ containing x ℓ . In particular, the vertices in { u ′ ℓ − , v ′ ℓ − , u ′ ℓ +1 , v ′ ℓ +1 , x ′ ℓ } are pairwise distinct and d ≥
5. See Fig. 8 ( a ) and ( b ) for two drawings of this situation.Let t , t , . . . , t d − denote the vertices in N ( x ℓ ) − { y ℓ − , x ℓ +1 , u ℓ , v ℓ } ; see Fig. 8 ( c ). Consider a path Q i = t i − y ′ ℓ − . Recall that Q i has length at most D −
1. Since x ℓ cannot be contained in a further short cycle, Q i must be a ( D − y ′ ℓ − not contained in { u ′ ℓ − , v ′ ℓ − , u ′ ℓ +1 , v ′ ℓ +1 , x ′ ℓ } .Therefore, we have that d ≥ Q r and Q s containing a common neighbour of y ′ ℓ − . This way, x ℓ would be contained in a third short cycle, acontradiction.As a result, we conclude that the repeat graph Υ of C is indeed a (2 D − C ′ as claimed. Thiscompletes the proof of Claim 1, and thus, of the lemma. ✷ While not of primary interest, it is not difficult to prove now that the cycles C , C , . . . , C k in theprevious lemma are pairwise disjoint.We call the aforementioned cycle C ′ the repeat of the cycle C in Γ, and denote it by rep( C ). Next somesimple consequences of the Repeat Cycle Lemma follow. Corollary 4.2 (Repeat Cycle Uniqueness)
If a short cycle C has a repeat cycle C ′ then C ′ is unique. C ℓ − C ℓ x ℓ +1 y ℓ = x ℓ C ℓ +1 ... ... z = x ′ ℓ = y ′ ℓ y ℓ − v ′ ℓ − x ′ ℓ +1 = y ′ ℓ − u ′ ℓ − x ′ ℓ +1 = y ′ ℓ − v ′ ℓ +1 u ′ ℓ +1 u ℓ v ℓ CC ℓ − C ℓ y ℓ = x ℓ ... ... y ℓ − v ′ ℓ − u ′ ℓ − x ′ ℓ +1 = y ′ ℓ − v ′ ℓ +1 u ′ ℓ +1 u ℓ v ℓ ( a ) ( b ) CC ℓ − C ℓ x ℓ +1 y ℓ = x ℓ C ℓ +1 ... ... z = x ′ ℓ = y ′ ℓ y ℓ − v ′ ℓ − x ′ ℓ +1 = y ′ ℓ − u ′ ℓ − x ′ ℓ +1 = y ′ ℓ − v ′ ℓ +1 u ′ ℓ +1 u ℓ v ℓ ... t t d − ( c ) C ℓ +1 x ℓ +1 z Figure 8: Auxiliary figure for Claim 1 of Lemma 4.2.
Corollary 4.3 (Repeat Cycle Symmetry) If C ′ = rep( C ) then C = rep( C ′ ) . Corollary 4.4
Let C and C be two short cycles in a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ which intersect at a path I of length smaller than D − , and let I ′ = rep C ( I ) . Then, the repeatcycle of C intersects C at I ′ . Corollary 4.5 (Handy Corollary)
Let C be a short cycle in a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ , and x, x ′ repeat vertices in C . Let C and C be the other short cycles containing x and x ′ , respectively. Suppose that I = C ∩ C is a path of length smaller than D − . Then, setting y = rep C ( x ) and y ′ = rep C ( x ′ ) , we have that y and y ′ are repeat vertices in the repeat cycle of C . roof. We denote the k neighbour cycles of C as E , E , . . . E k and their respective intersection pathswith C as I = x − y , I = x − y , . . . , I k = x k − y k in such way that C = x I y x I y . . . x k I k y k x . For1 ≤ j ≤ k , we also denote I ′ j = x ′ j − y ′ j , where x ′ j = rep E j ( x j ) and y ′ j = rep E j ( y j ).Obviously, for some r, s (1 ≤ r, s ≤ k ) we have that C = E r , C = E s , x ∈ I r , x ′ ∈ I s , y ∈ I ′ r , and y ′ ∈ I ′ s . We may assume r < s . By the Repeat Cycle Lemma, the vertices y and y ′ be-long to the repeat cycle C ′ of C . Then the paths xI r y r x r +1 I r +1 y r +1 . . . x s − I s − y s − x s I s x ′ ⊂ C and yI ′ r y ′ r x ′ r +1 I ′ r +1 y ′ r +1 . . . x ′ s − I ′ s − y ′ s − x ′ s I ′ s y ′ ⊂ C ′ are both ( D − ✷ Proposition 4.2
The set S (Γ) of short cycles in a bipartite ( d, D, − -graph Γ with d ≥ and D ≥ can be partitioned into sets S D − (Γ) , S D − (Γ) and S D − (Γ) , where S D − (Γ) is the set of short cycles whose intersections with neighbour cycles are ( D − -paths; S D − (Γ) is the set of short cycles whose intersections with neighbour cycles are ( D − -paths; and S D − (Γ) is the set of short cycles whose intersections with neighbour cycles are paths of length atmost D − . Proof.
If Γ is one of the non-regular graphs in Fig. 2 the result trivially follows. We then assume that Γis regular.Let C be a short cycle in Γ. If C is contained in a Θ D − then, according to Proposition 4.1, all theintersections of C with its neighbour cycles are ( D − C ∈ S D − (Γ).Now suppose that, for some short cycle C , P = C ∩ C is a path of length D −
2. Note that allvertices in P are saturated. Let v be an arbitrary vertex in P , v ′ = rep C ( v ), and C the short cycleother than C containing v ′ . Suppose that P = C ∩ C is not a ( D − P cannot be a( D − D −
3. But according to Corollary 4.4, the cycle rep ( C ) intersects C at exactly rep C ( P ), a proper subpath of P . This implies that rep ( C ) is a third short cycle containingthe vertex v , a contradiction. Consequently, the intersections of C with its (exactly two) neighbour cyclesare ( D − C ∈ S D − (Γ).Finally, if there is a short cycle intersecting C at a path of length at most D − C with all of its neighbour cycles are paths of length at most D −
3, and C ∈ S D − (Γ). ✷ The preceding result could be stated alternatively in term of vertices as follows:
Proposition 4.3
The set V (Γ) of vertices in a regular bipartite ( d, D, − -graph Γ with d ≥ and D ≥ can be partitioned into sets V D − (Γ) , V D − (Γ) and V D − (Γ) , where D − (Γ) is the set of vertices contained in cycles of S D − (Γ) ; V D − (Γ) is the set of vertices contained in cycles of S D − (Γ) ; V D − (Γ) is the set of vertices contained in cycles of S D − (Γ) ;and S D − (Γ) , S D − (Γ) , S D − (Γ) are defined as in Proposition 4.2. ✷ Θ D − Theorem 5.1
A bipartite ( d, D, − -graph Γ with d ≥ and D ≥ does not contain a subgraph isomor-phic to Θ D − . Proof.
Suppose that Γ has a subgraph Θ isomorphic to Θ D − , with branch vertices a and b . Let p , p , p , p and p be as in Fig. 9 ( a ), and let q be one of the neighbours of p not contained in Θ.Since all vertices of Θ are saturated, there cannot be a short cycle in Γ containing any of the incidentedges of p , p , p , p or p which are not contained in Θ. According to this and by applying the SaturatingLemma, there is an additional short cycle D in Γ such that q and one of the neighbours of p notcontained in Θ (say q ) are repeats in D , and D ∩ Θ = ∅ . Analogously, in Γ there is an additional shortcycle D such that q and one of the neighbours of p not contained in Θ (say q ) are repeats in D , and D ∩ Θ = ∅ ; an additional short cycle D such that q and one of the neighbours of p not contained in Θ(say q ) are repeats in D , and D ∩ Θ = ∅ ; and an additional short cycle D such that q and one of theneighbours of p not contained in Θ (say q ) are repeats in D , and D ∩ Θ = ∅ . See Fig. 9 ( b ).Note that D ∩ D is a path of length at most 2 < D −
2; otherwise for some vertex t ∈ D ∩ D theclosed walk tD q p bp q D t would contain a cycle of length at most 2 D − b wouldbelong, a contradiction. For similar reasons, the intersection paths D ∩ D and D ∩ D all have lengthat most 2, with 2¡ D −
2. We now apply the Handy Corollary. By mapping the cycle D to C , the vertex q to x , the vertex q to x ′ , the cycle D to C , the cycle D to C , the vertex q to y and the vertex q to y ′ , we obtain that q and q are repeat vertices in the repeat cycle of D . Therefore, since q ∈ D , itfollows that D and D are repeat cycles and q = q . This way, there would be a cycle q p bp q in Γ oflength 4 < D − D ≥ D − ✷ Proposition 5.1
The number N D − of short cycles in a bipartite ( d, D, − -graph Γ with d ≥ and D ≥ is given by the expression × (cid:0) d − ... +( d − D − (cid:1) − D − . D D D a bp p q q p p q q p q a bp p p p q p ( a ) ( b )Θ Θ Figure 9: Auxiliary figure for Theorem 5.1
Proof.
By Theorem 5.1, Γ does not contain a subgraph isomorphic to Θ D − . Then, according to Propo-sition 4.1, every vertex of Γ is contained in exactly two short cycles. We then count the number N D − ofshort cycles of Γ . Since the order of Γ is 2 × (cid:0) d −
1) + . . . + ( d − D − (cid:1) −
4, we have that N D − = × (cid:16) × (cid:0) d − ... +( d − D − (cid:1) − (cid:17) D − = × (cid:0) d − ... +( d − D − (cid:1) − D − ,and the proposition follows. ✷ ( d, D, − -graphs Since the number of short cycles in a graph Γ must be an integer, the expression obtained for N D − in Proposition 5.1 already suffices to prove the non-existence of bipartite ( d, D, − d, D ).Consider first the case in which D − p q is an odd prime power. Let G = { , , . . . , p − } be themultiplicative group of the field Z /p Z , let d − , p ), and let H be the cyclic subgroup of G generated by d −
1. We observe that the sum of the elements of H is null (mod p ). Furthermore, sincethe order of H divides the order of G , it must also divide p q − D −
2. Thus, we have2 × (cid:0) d −
1) + . . . + ( d − D − (cid:1) − ≡ − p ) if d − ≡ , p ),2( d − − p ) if d − , p ).Therefore, it immediately follows 16 orollary 5.1 There is no bipartite ( d, D, − -graph with d ≥ and D ≥ such that D − is an oddprime power. ✷ More generally, if p is an odd prime factor of D − D − ≡ r (mod p − × (cid:0) d −
1) + . . . + ( d − D − (cid:1) − ≡ − p ) if d − ≡ , p ),2 ( d − r +1 − d − − p ) if d − , p ); Corollary 5.2
There is no bipartite ( d, D, − -graph with d ≥ and D ≥ such that d − ≡ , p ) , where p is an odd prime factor of D − . ✷ It is also possible to examine completely the case of some small odd prime factors of D −
1. For example,it is not difficult to verify that, if D − k then 3 does not divide 2 × (cid:0) d −
1) + . . . + ( d − D − (cid:1) − Corollary 5.3
There is no bipartite ( d, D, − -graph with d ≥ and D ≥ such that D − ≡ . Now we turn to structural arguments to obtain other non-existence results.
Lemma 5.1
Any two non-disjoint short cycles in a bipartite ( d, D, − -graph Γ with d ≥ and D ≥ intersect at a path of length smaller than D − . Proof.
Since Γ does not contain a graph isomorphic to Θ D − , it is only necessary to prove here that anytwo non-disjoint short cycles in Γ cannot intersect at a path of length D − C and C in Γ intersecting at a path I of length D −
2. According to Proposition 4.2, C is intersected by exactly two short cycles, namely C and C , at two independent ( D − C , C , C , . . . , C m of pairwise distinct short cycles inΓ such that C i intersects C i +1 at a path I i of length D − ≤ i ≤ m − C i ∩ C j = ∅ for any i, j ∈ { , . . . , m } such that 2 ≤ | i − j | ≤ m − I = x − y , . . . , I m − = x m − − y m − in such way that, for 1 ≤ i ≤ m − x i ∼ x i +1 and y i ∼ y i +1 are edges in Γ. Also, let x ∈ N ( x ) ∩ ( C − I ), y ∈ N ( y ) ∩ ( C − I ), x m ∈ N ( x m − ) ∩ ( C m − I m − ), and y m ∈ N ( y m − ) ∩ ( C m − I m − ); see Fig. 10 ( a ). Set I = x − y and I m = x m − y m . Since the sequence C , C , C , . . . , C m is maximal and all the vertices in I , . . . , I m − aresaturated, it follows that I = I m , and we have either x = x m and y = y m (Fig. 10 ( b )), or x = y m and y = x m (Fig. 10 ( c )). 17 a ) x x x x m − x m − x m y y y y y m − y m − y m C C C C m − C m I I I I m − I m − I m I x = x m x x x x x x m − y y y y y y m − y = y m ( b ) . . .. . .. . . x = y m x x x x x y y y y y y m − y = x m ( c ) C C C C C C C C C C ... ... D − D − D − D − D − D − D − I I I I I I = I m I I I I I I m − I = I m x m − I m − x Figure 10: Auxiliary figure for Lemma 5.1If x = x m and y = y m , then m ≥ D ; otherwise the cycle x x . . . x m x would have length at most2 D −
2, contradicting the saturation of x . If conversely x = y m and y = x m then m ≥ D ; otherwisethe cycle x x . . . x m y y . . . y m x containing x would have length at most 2 D −
2, a contradiction as well.For our purposes, it is enough to state m ≥ D ≥ p be the neighbour of y on I , and p i +1 = rep C i +1 ( p i ) for 1 ≤ i ≤
4. Also, let q be a neighbourof p not contained in I ; see Fig. 11 ( a ).Since all vertices on I are saturated, the edge q ∼ p cannot be contained in a further short cycle.We apply the Saturating Lemma (by mapping C to C , p to α , p to α ′ , and q to γ ), and obtain in Γan additional short cycle D such that q and one of the neighbours of p not contained in I (say q ) arerepeats in D , and D ∩ C = ∅ . Analogously, for 2 ≤ i ≤ D i in Γ18uch that q i and a neighbour of p i +1 not contained in I i +1 (say q i +1 ) are repeats in D i , and D i ∩ C i +1 = ∅ ;see Fig. 11 ( b ). . . .. . . . . .. . .x x x x x y y y y y p p p p p q C C C C . . .. . . . . .. . .x x x x x y y y y y p p p p p q q q q q D D D D C C C C ( a )( b ) D − D − D − D − D − Figure 11: Auxiliary figure for Lemma 5.1For i = 1 or 3, D i ∩ D i +1 cannot be a ( D − t i ∈ D i ∩ D i +1 , therewould be a cycle q i p i y i y i +1 y i +2 p i +2 q i +2 D i +1 t i D i q i of length at most 6 + D − D − D − ≥ p i is saturated and g(Γ) = 2 D −
2. Analogously, D ∩ D cannot be a( D − D to C , D to C and D to C , and thevertices q to x , q to x ′ , q to y , and q to y ′ , it follows that the vertices q and q are repeat vertices inthe repeat cycle of D . Since q ∈ D , D and D are repeat cycles and q = q . This way, we obtain acycle q p y y y y y p q in Γ of length 8 < D −
2, a contradiction.This completes the proof of the lemma. ✷ Theorem 5.2
There are no bipartite ( d, D, − -graphs for d ≥ and odd D ≥ . roof. The case D = 5 can be easily discarded by using Proposition 5.1, so we assume D ≥ d, D, − d ≥ D ≥
7. According to Lemma 5.1,any two non-disjoint short cycles in Γ intersect at a path of length smaller than D −
2, which means thatevery short cycle C in Γ has a repeat cycle C ′ (by the Repeat Cycle Lemma). Because of the uniquenessand symmetry of repeat cycles, the number N D − of short cycles in Γ must be even.However, since D is odd, the number N D − = × (cid:0) d − ... +( d − D − (cid:1) − D − of short cycles in Γ is odd, acontradiction. ✷ Furthermore, using Theorem 5.2 and Proposition 5.1 we complete the catalogue of bipartite ( d, D, − ≤ D ≤ D ≥
5, the only bipartite (2 , D, − Theorem 5.3
The path of length 5 is the only bipartite (∆ , D, − -graph with ∆ ≥ and ≤ D ≤ . (3 , D, − -graphs with D ≥ In this section we complete the catalogue of bipartite (3 , D, − , D, − D ≥ Lemma 5.2
Any two non-disjoint short cycles in a bipartite (3 , D, − -graph Γ with d ≥ and D ≥ intersect at a path of length smaller than D − . Proof.
By Lemma 5.1, it is only necessary to prove here that any two short cycles C and C in Γ cannotintersect at a path I = x − y of length D −
3. We proceed by contradiction. Let x ′ and y ′ be therepeat vertices of x and y in C , respectively. By Corollary 4.4, the repeat cycle C ′ of C intersects C at I ′ = x ′ − y ′ (the repeat path of I in C ); see Fig. 12. If we denote by z the neighbour of x on C − C , thenwe have that the other short cycle containing z would also contain at least one of the vertices in { x, y ′ } ,which contradicts the fact that x and y ′ are both saturated. ✷ Theorem 5.4
There are no bipartite (3 , D, − -graphs with even D ≥ . Proof.
Recall that Theorem 5.3 covers the case D = 6.Let Γ be a bipartite (3 , D, − D ≥ C a short cycle in Γ, and x , x ′ two repeatvertices in C . Let x and x ′ be the neighbours of x and x ′ , respectively, not contained in C . According tothe Saturating Lemma, there is an additional short cycle C containing x and x ′ such that C ∩ C = ∅ .Let y be one of the neighbours of x contained in C , and y ′ = rep C ( y ). Denote by x and x ′ the20 C C ′ I I ′ x y ′ zy x ′ D − D − Figure 12: Auxiliary figure for Lemma 5.2neighbours of y and y ′ , respectively, not contained in C . Again by the Saturating Lemma, there is anadditional short cycle C such that x ′ = rep C ( x ) and C ∩ C = ∅ . Since d = 3, we may assume thatthe other short cycle C containing x also contains x and a neighbour of x in C (say y ). We first provethat C ∩ C = y x . x x x x ′ x ′ x ′ y z y y ′ y y ′ z ′ C C C x x x x ′ x ′ x ′ y z y y ′ y y ′ z ′ C C C P ( a ) ( b ) x ′ y ′ x C D − Figure 13: Auxiliary figure for Theorem 5.4.
Claim 1. C ∩ C = y x . Proof of Claim 1.
Let y , z , y , y ′ and z ′ be as in Fig. 13 ( a ).Consider a path P = x ′ − y . If y ′ ∈ P , then P would go through a neighbour of y ′ contained in C and there would be a cycle in Γ of length at most 2 D −
4. Therefore, we may assume y ′ ∈ P . If { x ′ , y ′ , z ′ } ⊂ V ( P ∩ C
2) then there would be a short cycle intersecting the cycle C at a path of length D −
3, a contradiction to Lemma 5.2. Similarly, we have that z P . Also, P must be a ( D − x P ; otherwise there would be a short cycle intersecting the cycle C at a path of length D −
2, acontradiction to Lemma 5.1. 21 x x x y y y y y ( D ) x ( D ) x ( D +1) x ′ ( D +1) y ′ ( D +1) z ′ ( D +1) x ′ ( D ) y ′ ( D ) x ′ x ′ x ′ x ′ y ′ y ′ y ′ C C C C C ( D ) C ( D +1) z ′ z ′ z ′ ( D ) . . . ... . . . ... . . .. . . Figure 14: Auxiliary figure showing the sequence C , C , . . . , C D/ of short cycles.For 3 ≤ i ≤ D/ x i and x ′ i be the neighbours of y i − and y ′ i − , respectively, not containedin C i − , and let C i be (in virtue of the Saturating Lemma) the additional short cycle disjoint from C i − which contains x i and x ′ i . Since P must go through x ′ i , we denote by y ′ i the neighbour of x ′ i on P ∩ C i and set y i = rep C i ( y ′ i ). We now show that, if i = D/ P ∩ C i = x ′ i y ′ i . Assume the contrary; thatis, P ∩ C i = x ′ i y ′ i z ′ i (since g(Γ) = 2 D − | V ( P ∩ C i ) | ≤ y P z ′ i C i x i y i − x i − y i − x i − . . . y x x y intersecting C i at a path of length D −
3, contradicting Lemma 5.2(see Figures 13 ( b ) and 14). Consequently, P ∩ C i = x ′ i y ′ i and P must go through a neighbour of y ′ i notcontained in C i .This way, for 3 ≤ i ≤ D/ d ( y , y ′ i ) = d ( y , y ′ i − ) − D − i − d ( y , y ′ D/ ) = 0. Since the cycle C D/ contains the vertices y ∈ C ∩ C ∩ P and x ′ D/ ∈ P − C , wehave C D/ = C , which implies that C ∩ C = y x . ✷ As the selection of C and C was arbitrary, basically as a corollary of Claim 1 we have: Claim 2.
Any two non-disjoint short cycles in Γ intersect at an edge.Finally, suppose that C and C intersect at y x , as stated by Claim 1. Let y ′ be the repeat vertexof y in C ; then, by Corollary 4.4, the repeat cycle C ′ of C intersects C at y ′ x ′ (the repeat path of y x in C ). Setting Q = x C y ′ = x w . . . w D − y ′ , we have Q is a path of length D − F , . . . , F D/ − of short cycles22uch that F i ∩ C = w i − w i . However, since D is even, the other short cycle containing w D − would alsocontain one of the vertices in { w D − , y ′ } , which contradicts the fact that w D − and y ′ are both saturated.This completes the proof of Theorem 5.4. ✷ C x y C ′ C x ′ y ′ w w w D − Figure 15: Auxiliary figure for Theorem 5.4.Combining Theorems 5.2 and 5.4, we have that the only bipartite (3 , D, − D ≥ The main results obtained in this paper are summarised below.First we stated important structural properties of bipartite ( d, D, − d ≥ D ≥ d, D, − d ≥ D ≥ d, D ); this includedthe case in which D − D − ≡ d, D, − d ≥ D ≥ , D, − ≥ ≤ D ≤ , D, − ǫ )-graphs with ∆ ≥
2, 5 ≤ D ≤ D = 6 and 0 ≤ ǫ ≤ Catalogue of bipartite (∆ , D, -graphs with ∆ ≥ and ≤ D ≤ . For 5 ≤ D ≤
187 and ∆ = 2the only Moore bipartite graphs are the 2 D -cycles, whereas for D = 6 and ∆ ≥ ≤ D ≤
187 and ∆ ≥ Catalogue of bipartite (∆ , D, − -graphs with ∆ ≥ and ≤ D ≤ . The results of [14] combinedwith [5, 6] showed that there are no such graphs.
Catalogue of bipartite (∆ , D, − -graphs with ∆ ≥ and ≤ D ≤ . The path of length 5 is theonly such graph. 23nother important result of the paper is the completion of the catalogue of bipartite (3 , D, − ǫ )-graphswith D ≥ ≤ ǫ ≤ Catalogue of bipartite (3 , D, -graphs with D ≥ . The cubic Moore bipartite graphs are the com-plete bipartite graph K , for D = 2, the unique incidence graph of the projective plane of order 2for D = 3, the unique incidence graph of the generalised quadrangle of order 2 for D = 4, and theunique incidence graph of the generalised hexagon of order 2 for D = 6. Catalogue of bipartite (3 , D, − -graphs with D ≥ . There are only two non-isomorphic (3 , D, − D ≥
2; a unique bipartite (3 , , − , , − a ). Catalogue of bipartite (3 , D, − -graphs with D ≥ . There exist no bipartite (3 , , − , , − , , − , D, − D ≥
5, outcome that gives an alternative proof of the optimality of the known bipartite(3 , , − ( d, D, − -graphs with d ≥ and D = 3 , The main results in this paper did not include bipartite ( d, D, − d ≥ D = 3 , , , − , , − d, D, − d ≥ D = 3 ,
4. With our current approach we cannot have Theorem 5.1 for D = 3 , D ∩ D , D ∩ D and D ∩ D have length at most 2, and for usto apply the the Repeat Cycle Lemma we need the lengths of such paths to be less than D −
2. Indeed,the graph in Fig. 4 ( a ) offers a good illustration of this. Even if we had Theorem 5.1, something similarwould occur with Lemma 5.1; see the graphs in Fig. 4 ( b ) and Fig. 5.24 .2 Remarks on the upper bound for N b (∆ , D ) Our results improve the upper bound on N b ∆ ,D for many combinations of ∆ and D . Recall that a bipartite(∆ , D, − ≥ D ≥ Proposition 6.1
For natural numbers ∆ ≥ and D ≥ such that D − is an odd prime power, N b (∆ , D ) ≤ M b (∆ , D ) − . Proposition 6.2
For natural numbers ∆ ≥ and D ≥ such that d − ≡ , p ) , where p is anodd prime factor of D − , N b (∆ , D ) ≤ M b (∆ , D ) − . Proposition 6.3
For natural numbers ∆ ≥ and D ≥ such that D ≡ , , , , N b (∆ , D ) ≤ M b (∆ , D ) − . Proposition 6.4
For natural numbers ∆ ≥ and ≤ D ≤ ( D = 6 ), N b (∆ , D ) ≤ M b (∆ , D ) − . Proposition 6.5
For any natural number D ≥ ( D = 6 ), N b (3 , D ) ≤ M b (3 , D ) − . Finally, we feel that the next conjectures are valid.
Conjecture 6.1
There is no bipartite (∆ , D, − -graph with ∆ ≥ and D ≥ . Conjecture 6.2
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