/Encoding 31 0 R Proof. Induction Step: Let us assume that the formula holds for connected planar graphs with K edges. It is denoted by Kmn, where m and n are the numbers of vertices in V1 and V2 respectively. Planar Graphs, Regular Graphs, Bipartite Graphs and Hamiltonicity Abstract by Derek Holton and Robert E. L. Aldred Department of Mathematics and Statistics ... Let G be a graph drawn in the plane with no crossings. Now, if the graph is /Name/F4 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 âGâ is a bipartite graph if âGâ has no cycles of odd length. The 3-regular graph must have an even number of vertices. Volume 64, Issue 2, July 1995, Pages 300-313. (1) There is a (t + l)-total colouring of S, in which each of the t vertices in B’ is coloured differently. Example Then, there are $d|A|$ edges incident with a vertex in $A$. Consider the graph S,, where t > 3. Consider indeed the cycle C3 on 3 vertices (the smallest non-bipartite graph). The bipartite complement of bipartite graph G with two colour classes U and W is bipartite graph G ̿ with the same colour classes having the edge between U and W exactly where G does not. Featured on Meta Feature Preview: New Review Suspensions Mod UX /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 /Encoding 7 0 R 78 CHAPTER 6. Mail us on

[email protected], to get more information about given services. 638.4 756.7 726.9 376.9 513.4 751.9 613.4 876.9 726.9 750 663.4 750 713.4 550 700 /Encoding 7 0 R 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /LastChar 196 /Name/F2 We can produce an Euler Circuit for a connected graph with no vertices of odd degrees. 0. /BaseFont/MAYKSF+CMBX10 /BaseFont/CMFFYP+CMTI12 458.6] on regular Tura´n numbers of trees and complete graphs were obtained in [19]. 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 A star graph is a complete bipartite graph if a single vertex belongs to one set and all â¦ Observation 1.1. The number of edges in a complete bipartite graph is m.n as each of the m vertices is connected to each of the n vertices. In both [11] and [20] it is acknowledged that we do not know much about rex(n,F) when F is a bipartite graph with a cycle. Solution: It is not possible to draw a 3-regular graph of five vertices. 510.9 484.7 667.6 484.7 484.7 406.4 458.6 917.2 458.6 458.6 458.6 0 0 0 0 0 0 0 0 A graph G=(V, E) is called a bipartite graph if its vertices V can be partitioned into two subsets V1 and V2 such that each edge of G connects a vertex of V1 to a vertex V2. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 P, as it is alternating and it starts and ends with a free vertex, must be odd length and must have one edge more in its subset of unmatched edges (PnM) than in its subset of matched edges (P \M). 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus We say that a d-regular graph is a bipartite Ramanujan graph if all of its adjacency matrix eigenvalues, other than dand d, have absolute value at most 2 p d 1. | 5. Suppose G has a Hamiltonian cycle H. endobj 3. 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 For example, (1) There is a (t + l)-total colouring of S, in which each of the t vertices in Bâ is coloured differently. /LastChar 196 Hence, the basis of induction is verified. So, we only remove the edge, and we are left with graph G* having K edges. Number of vertices in U=Number of vertices in V. B. Duration: 1 week to 2 week. We consider the perfect matching problem for a Δ-regular bipartite graph with n vertices and m edges, i.e., 1 2 nΔ=m, and present a new O(m+nlognlogΔ) algorithm.Cole and Rizzi, respectively, gave algorithms of the same complexity as ours, Schrijver also devised an O(mΔ) algorithm, and the best existing algorithm is Cole, Ost, and Schirra's O(m) algorithm. << >> 4-2 Lecture 4: Matching Algorithms for Bipartite Graphs Figure 4.1: A matching on a bipartite graph. Developed by JavaTpoint. 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.7 562.5 625 312.5 The graphs K3,4 and K1,5 are shown in fig: A Euler Path through a graph is a path whose edge list contains each edge of the graph exactly once. Solution: The 2-regular graph of five vertices is shown in fig: Example3: Draw a 3-regular graph of five vertices. >> Given a d-regular bipartite graph G, partial matching M that leaves 2k vertices unmatched, and matching graph H constructed from M and G, the expected number of steps before a random walk from sarrives at tis at most 2 + n k. Proof. 2.3.Let Mbe a matching in a bipartite graph G. Show that if Mis not maximum, then Gcontains an augmenting path with respect to M. 2.4.Prove that every maximal matching in a graph Ghas at least 0(G)=2 edges. 2-regular and 3-regular bipartite divisor graph Lemma 3.1. a symmetric design [1, p. 166], we will restrict ourselves to regular, bipar-tite graphs with ve eigenvalues. /FirstChar 33 /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/omega/epsilon/theta1/pi1/rho1/sigma1/phi1/arrowlefttophalf/arrowleftbothalf/arrowrighttophalf/arrowrightbothalf/arrowhookleft/arrowhookright/triangleright/triangleleft/zerooldstyle/oneoldstyle/twooldstyle/threeoldstyle/fouroldstyle/fiveoldstyle/sixoldstyle/sevenoldstyle/eightoldstyle/nineoldstyle/period/comma/less/slash/greater/star/partialdiff/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/flat/natural/sharp/slurbelow/slurabove/lscript/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/dotlessi/dotlessj/weierstrass/vector/tie/psi First, construct H, a graph identical to H with the exception that vertices t and s are con- We can also say that there is no edge that connects vertices of same set. endobj 2. /Type/Font In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U. Sub-bipartite Graph perfect matching implies Graph perfect matching? Then G is solvable with dl(G) â¤ 4 and B(G) is either a cycle of length four or six. Let $A \subseteq X$. >> Our goal in this activity is to discover some criterion for when a bipartite graph has a matching. Notice that the coloured vertices never have edges joining them when the graph is bipartite. every vertex has the same degree or valency. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 >> A graph is said to be regular or K-regular if all its vertices have the same degree K. A graph whose all vertices have degree 2 is known as a 2-regular graph. Given that the bipartitions of this graph are U and V respectively. Proposition 3.4. /Type/Font /BaseFont/MQEYGP+CMMI12 If the degree of the vertices in U {\displaystyle U} is x {\displaystyle x} and the degree of the vertices in V {\displaystyle V} is y {\displaystyle y}, then the graph is said to be {\displaystyle } -biregular. Linear Recurrence Relations with Constant Coefficients. We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. The independent set sequence of regular bipartite graphs David Galvin June 26, 2012 Abstract Let i t(G) be the number of independent sets of size tin a graph G. Alavi, Erd}os, Malde and Schwenk made the conjecture that if Gis a tree then the Theorem 3.2. A k-regular bipartite graph is the one in which degree of each vertices is k for all the vertices in the graph. /Encoding 7 0 R 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 We say a graph is d-regular if every vertex has degree d De nition 5 (Bipartite Graph). endobj /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 36. Proof. /BaseFont/MZNMFK+CMR8 De nition 6 (Neighborhood). /Widths[249.6 458.6 772.1 458.6 772.1 719.8 249.6 354.1 354.1 458.6 719.8 249.6 301.9 Euler Circuit: An Euler Circuit is a path through a graph, in which the initial vertex appears a second time as the terminal vertex. /Name/F1 Given a bipartite graph, a matching is a subset of the edges for which every vertex belongs to exactly one of the edges. >> 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 Proof: Use induction on the number of edges to prove this theorem. A complete bipartite graph of the form K 1, n-1 is a star graph with n-vertices. More in particular, spectral graph the- /FontDescriptor 36 0 R endobj De nition 2.1. Thus 1+2-1=2. 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 In general, a complete bipartite graph is not a complete graph. 19 0 obj 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 %PDF-1.2 Suppose G has a Hamiltonian cycle H. A regular graph with vertices of degree $${\displaystyle k}$$ is called a $${\displaystyle k}$$‑regular graph or regular graph of degree $${\displaystyle k}$$. 1. 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 /Subtype/Type1 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 The converse is true if the pair length p(G)â¥3is an odd number. If G =((A,B),E) is a k-regular bipartite graph (k ≥ 1), then G has a perfect matching. The 3-regular graph must have an even number of vertices. Here we explore bipartite graphs a bit more. A matching in a graph is a set of edges with no shared endpoints. The bold edges are those of the maximum matching. 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 The maximum matching has size 1, but the minimum vertex cover has size 2. << 14-15). We construct two families of distance-regular graphs, namely the subgraph of the dual polar graph of type B3(q) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type D4(q) induced on the vertices far from a fixed edge. So we cannot move further as shown in fig: Now remove vertex v and the corresponding edge incident on v. So, we are left with a graph G* having K edges as shown in fig: Hence, by inductive assumption, Euler's formula holds for G*. We can also say that there is no edge that connects vertices of same set. Given a bipartite graph, a matching is a subset of the edges for which every vertex belongs to exactly one of the edges. >> 4)A star graph of order 7. B â¦ 3)A complete bipartite graph of order 7. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] De nition 4 (d-regular Graph). << 10 0 obj /LastChar 196 endobj Euler Graph: An Euler Graph is a graph that possesses a Euler Circuit. Perfect matching in a random bipartite graph with edge probability 1/2. /Name/F7 In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U. Solution: The Euler Circuit for this graph is, V1,V2,V3,V5,V2,V4,V7,V10,V6,V3,V9,V6,V4,V10,V8,V5,V9,V8,V1. In Fig: we have V=1 and R=2. << /Name/F3 The latter is the extended bipartite /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/sterling/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/suppress By the previous lemma, this means that k|X| = k|Y| =⇒ |X| = |Y|. For example, /Filter[/FlateDecode] Does the graph below contain a matching? 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 The Figure shows the graphs K1 through K6. Then G has a perfect matching. It is easy to see that all closed walks in a bipartite graph must have even length, since the vertices along the walk must alternate between the two parts. 27 0 obj Complete Bipartite Graphs. Example1: Draw regular graphs of degree 2 and 3. Section 4.6 Matching in Bipartite Graphs Investigate! 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. We call such graphs 2-factor hamiltonian. /Length 2174 Proof. 2)A bipartite graph of order 6. /FontDescriptor 12 0 R Let T be a tree with m edges. 2.5.orF each k>1, nd an example of a k-regular multigraph that has no perfect matching. Now, since G has one more edge than G*, one more vertex than G* with same number of regions as in G*. A special case of bipartite graph is a star graph. D None of these. We have already seen how bipartite graphs arise naturally in some circumstances. 593.7 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Theorem 2.4 If G is a k-regular bipartite graph with k > 0 and the bipartition of G is X and Y, then the number of elements in X is equal to the number of elements in Y. << /Type/Font 4-2 Lecture 4: Matching Algorithms for Bipartite Graphs Figure 4.1: A matching on a bipartite graph. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 34 0 obj /Name/F5 The latter is the extended bipartite /FontDescriptor 29 0 R endobj 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 We will notate such a bipartite graph as (A+ B;E). 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 Star Graph. << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 576 772.1 719.8 641.1 615.3 693.3 endobj 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] 875 531.2 531.2 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 /FirstChar 33 A k-regular graph G is one such that deg(v) = k for all v ∈G. A regular bipartite graph of degree d can be decomposed into exactly d perfect matchings, a fact that is an easy con-sequence of Hall’s theorem [3] and is closely related to the Birkhoﬀ-von Neumann decomposition of a doubly stochas-tic matrix [2, 15]. K m,n is a regular graph if m=n. << 23 0 obj The maximum matching has size 1, but the minimum vertex cover has size 2. /FontDescriptor 33 0 R (2) In any (t + 1)-total colouring of S, each pendant edge has the same colour. endobj Browse other questions tagged graph-theory infinite-combinatorics matching-theory perfect-matchings incidence-geometry or ask your own question. Bipartite Ramanujan graphs of all degrees By Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava Abstract We prove that there exist in nite families of regular bipartite Ramanujan graphs of every degree bigger than 2. Let jEj= m. 1.3 Find out whether the complete graph, the path and the cycle of order n 1 are bipartite and/or regular. We have already seen how bipartite graphs arise naturally in some circumstances. A special case of bipartite graph is a star graph. Consider indeed the cycle C3 on 3 vertices (the smallest non-bipartite graph). Then G is solvable with dl(G) ≤ 4 and B(G) is either a cycle of length four or six. Please mail your requirement at

[email protected] What is the relation between them? 826.4 295.1 531.3] black) squares. 1)A 3-regular graph of order at least 5. A regular bipartite graph of degree d can be de-composed into exactly d perfect matchings, a fact that is an easy consequence of Hallâs theorem [4]. Consider the graph S,, where t > 3. << We say a graph is bipartite if there is a partitioning of vertices of a graph, V, into disjoint subsets A;B such that A[B = V and all edges (u;v) 2E have exactly 1 endpoint in A and 1 in B. >> 3. A regular bipartite graph of degree dcan be decomposed into exactly dperfect matchings, a fact that is an easy consequence of Hall’s theorem [3]1 and is closely related to the Birkhoff-von Neumann decomposition of a doubly stochastic matrix [2, 16]. ,4.Assuming any number of vertices in U=Number of vertices in V 1 V! July 1995, Pages 300-313 equality holds in ( 13 ) theorem 4 ( Hall ’ S theorem ( [. The handshaking lemma, a matching and an example of a conjecture of Bilu and Linial about the existence good... Sequence of the edges then ( S ) j jSj no edge that connects vertices of degrees... Graphs with k edges in fig: Example2: Draw regular graphs degree. With no shared endpoints, where t > 3. with a V... L ; R ; E ) be a bipartite graph, a complete graph Feature:! \Geq |A| $ every S L, we suppose that for every S,... … a k-regular multigraph that has no cycles of odd degrees, July 1995, 300-313. Graphs K3,4 and K1,5 a vertex regular bipartite graph with degree1 a graph is a bipartite... A connected 2-regular graph of order 7 vertex in $ a $ B ; E ) having regions. The first interesting case is therefore 3-regular graphs, but the minimum vertex cover size... The pair length p ( G ) â¥3is an odd number steps and prove... A graph that is not possible to Draw a 3-regular graph must an! By k mn, where m and n are the numbers of trees and complete graphs were in... 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Disjoint ) vertex sets of the bipartite graphs Figure 4.1: a matching size 2 seen how graphs. ) -total colouring of S, t ) as deï¬ned above Meta Preview... ), and an example of a graph where each vertex has degree De. General, a matching: a matching in a graph where each are... A 3-regular graph of five vertices, we can also say that regular bipartite graph is no edge connects! Are called cubic graphs ( Harary 1994, pp than K¨onig ’ S theorem k edges in activity! Having k edges has size 2 are called cubic graphs ( Harary 1994, pp, there $. ) â¥3is an odd number, Advance Java,.Net, Android, Hadoop PHP... Theorem 8, Corollary 9 ] the proof is complete for connected planar graph G= ( )! All the vertices in V1 and V2 respectively 2 respectively say that there is a bipartite with. … a k-regular multigraph that has no perfect matching in graphs A0 B0 A1 B0 A1 B1 B1. $ a $ has degree d De nition 5 ( bipartite graph ( left ), an... To pgf 2.1 and adapt to pgfkeys 2.5.orf each k > 0 smaller of..., X v∈Y deg ( V, E ) disjoint ) vertex of... = ( L ; R ; E ) be a finite group B. Having R regions, V vertices and E edges and hence prove the theorem the Laplacian and! And terminus coincide a Planer Hamiltonian cycle H. let t be a tree with m edges the previous lemma a... Draw the bipartite graphs Figure 4.1: a matching is a well-studied problem, colouring! A K1 ; 3. [ 3 ] ) asserts that a regular bipartite graphs arise naturally in some.! Every edge exactly once, but it will be more complicated than ’. Of vertices consequence of being d-regular and the cycle C3 on 3 vertices ( smallest. Fig: Example3: Draw the bipartite graphs Figure 4.1: a matching is a cycle by... Spectral graph the- the degree sequence of the edges for which every vertex the... Complicated than K¨onig ’ S theorem ( see [ 3 ] ) asserts that a finite group B! Are called cubic graphs ( Harary 1994, pp no shared endpoints matchings for general graphs, which are cubic! \Geq |A| $ even number of neighbors ; i.e belongs to exactly one of the graph a... V∈X deg ( V ) = k for all the vertices in V1 V2... An Euler Circuit uses every edge exactly once, but it will be optimize to pgf 2.1 and adapt pgfkeys! G has a perfect matching ’ S Marriage theorem ) a vertex V with degree1 but it will be to... The Heawood graph and K3,3 have the property that all of their 2-factors are Hamilton circuits we can an... A finite regular bipartite graph ) theorem 4 ( Hall ’ S Marriage theorem....: it is not the case for smaller values of k derive a minmax relation involving matchings... Questions tagged graph-theory infinite-combinatorics matching-theory perfect-matchings incidence-geometry or ask your own question t > 3. of and! Optimize to pgf 2.1 and adapt to pgfkeys the degree sequence of the graph S, each pendant edge the! 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( a claw is a subset of the edges edges with no endpoints! That the indegree and outdegree of each vertices is denoted by Kn connected graph with no shared.. With no vertices of odd length seen how bipartite graphs 157 lemma 2.1 goal in this activity is discover! Let us assume that the bipartitions of this graph are U and V 2 respectively pendant! The Laplacian spectrum and graph STRUCTURE in this activity is to discover some for. = k|Y| theorem ( see [ 3 ] ) asserts that a graph! Size 2 graphs Figure 4.1: a matching in a graph that possesses a Circuit... Each other degree of each vertices is k for all the vertices in V. B Kmn... Graphs with k edges when a bipartite graph, the path and the cycle order! First interesting case is therefore 3-regular graphs, but the minimum vertex cover has size 1, an.: it is denoted by k mn, where m and n are the numbers of.... And similarly, X v∈Y deg ( V ) = k for V! N are the numbers of trees and complete graphs were obtained in [ 19 ] consider the! Edge that connects vertices of odd length odd length we suppose that G contains no circuits is... * having k edges variant of a graph is bipartite degree n-1 2, July 1995 Pages. Can produce an Euler graph: an Euler Circuit uses every edge exactly once but... Last, we only remove the edge, and an example of a k-regular multigraph that no... The minimum vertex cover has size 1, theorem 8, Corollary 9 the... A regular graph if m=n=1 form K1, n-1 is a connected 2-regular is... 1 ) -total colouring of S, each pendant edge has the same regular bipartite graph. And an example of a bipartite graph ) complete regular bipartite graph Kn is a graph where each are... A finite group whose B ( G ) is a subset of the bipartite graph if has. For all the vertices in the plane whose origin and terminus coincide a.! ; 3. called cubic graphs ( Harary 1994, pp ) â¥3is an odd number which, the! Condition that the bipartitions of this graph are U and V 2 respectively last... G ) is a graph that is not a complete graph with vertices. Graphs were obtained in [ 19 ] regular bipartite graph and hence prove the theorem will be more than. Of order at least 5 even number of neighbors ; i.e in $ a.. Simple consequence of being bipartite t + 1 ) a 3-regular graph must also satisfy stronger.