4 regular graph

Volume 44, Issue 4. These include the Chvatal graph, Brinkmann graph (discovered independently by Kostochka), and Grunbaum graph. Answer: b For example, K is the smallest simple n 5 4-regular graph. Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains.. A 3-regular graph is known as a cubic graph.. A strongly regular graph is a regular graph … (c) What is the largest n such that Kn = Cn? We prove that each {claw, K4}-free 4-regular graph, with just one class of exceptions, is a line graph. As a matter of fact, I have encountered this family of 4-regular graphs, where every edges lies in exactly one C4, and no two C4 share more than one vertex. strongly regular. To get all such graphs this way, you need to start with any $4$-regular graph, take the line graph, and then carefully delete the matchings to avoid extra squares. Regular Graph. We conjecture that in this class even cycle decompositions always exists and prove the conjecture for cubic graphs with oddness at most 2. There are exactly one graph on 21 vertices and one on 25 vertices. In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. [7], Quartic graphs have an even number of Hamiltonian decompositions. According to Handshaking lemma:- [math]\displaystyle \sum_{v\ \epsilon\ V}deg\ v=2|E|[/math] Since degree of every vertices is 4, therefore sum of the degree of all vertices can be written as [math]N \times 4… There is a closed-form numerical solution you can use. This vector image was created with a text editor. Abstract. We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. Connected regular graphs with girth at least 7 . (e) Is Qn a regular graph for n ≥ 1? regular graph with parameters n 2 , n−2 2 , n−4 2 , n−3 2 . In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. If so, what is the degree of the vertices in Qn? [8], It is an open conjecture whether all quartic Hamiltonian graphs have an even number of Hamiltonian circuits, or have more than one Hamiltonian circuit. In other words, a quartic graph is a 4- regular graph. Several well-known graphs are quartic. 4. Date: 1 July 2016: Source: Own work: Author: xJaM: Other versions: Other two isomorphic such graphs are: The source code of this SVG is valid. $\begingroup$ The following easy construction provides a bunch of 4-regular graphs with each edge in a triangle: Start with a 3-regular graph. A graph G is said to be regular, if all its vertices have the same degree. (d) For what value of n is Q2 = Cn? In this note, we present a sequence of Hamiltonian 4-regular graphs whose domination numbers are sharp. Licensing . Connected regular graphs with girth at least 7 . Abstract. (a) How many edges are in K3,4? Is K5 a regular graph? In H.P.Tong-Viet (2013b), Hung P. Tong Viet studied the 3-regular graphs which might occur as prime graphs of some group G. In the same paper, he also conjectured that the only 4-regular graphs that can arise are the complete graph of order 5 and the 4-regular graph of order 6. $\endgroup$ – user67773 Jul 17 '14 at … This inequality, which must be true for every regular polyhedral graph, tells us about the possible values of n and d. First, notice that if n and d are both very large, then the left-hand side will be very small. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. Show that a regular bipartite graph with common degree at least 1 has a perfect matching. 4-regular graph 07 001.svg 435 × 435; 1 KB. Lovász conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among … 4-regular graph 07 001.svg 435 × 435; 1 KB. PDF | In this note we give the smallest 4-regular 4-chromatic graphs with girth 5. Regular Graph: 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. In this case, the boundary of its quadrilaterals Q is empty, because ever y edge is shared by two quadrilaterals. This forms the main agenda of our discussion. Regular Graph. The following table contains numbers of connected cubic graphs with given number of vertices and girth at least 7. Thomas Grüner found that there exist no 4-regular Graphs with girth 7 on less than 58 vertices. Explanation: In a regular graph, degrees of all the vertices are equal. A 4 regular graph on 6 vertices.PNG 430 × 331; 12 KB. For a 4-regular graph any 2-connected component must have an even number of edges, and the simplest of the conditions necessary for the existence of an ECD is always met if the graph has connectivity at least 2. A quartic graph is a graph which is 4- regular. For example, XC 1 represents W 4, gem. Describing what "carefully" entails, and deciding if it is even possible, may turn out to be difficult, though. To the best of my (M. DeVos') knowledge, this might be the full list of such graphs. However, in this paper, it is shown that the dual of a quadrilateral mesh on a 2-dimensional compact manifold with an even number of quadrilaterals (which is a 4-regular graph) always has a perfect matching. Digital-native fourth grade students are navigating an increasingly complex world. Media in category "4-regular graphs" The following 6 files are in this category, out of 6 total. A number of … When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. A trail (a closed walk with no edge repetition) in a graph is called a transverse path , or simply a transversal , if consecutive edges of the path are never … Regular Graph: 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. Note that 4 K is the smallest loopless 4-regular graph. It has an automorphism group of cardinality 72, and is referred to as d4reg9-14 below. In a simple graph, the number of edges is equal to twice the sum of the degrees of the vertices. Title: Decomposition of $(2k+1)$-regular graphs containing special spanning $2k$-regular Cayley graphs into paths of length $2k+1$ Authors: Fábio Botler , Luiz Hoffmann Download PDF If so, what is the degree of the vertices in Qn? When assumption (9) holds, dual of the graph is a 4-regular graph. https://doi.org/10.1016/j.disc.2011.12.007. (b) How many edges are in K5? A simple graph with ‘n’ vertices (n >= 3) and ‘n’ edges is called a cycle graph if all its … We also discuss even cycle double covers of cubic graphs. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with … Is K3,4 a regular graph? Our fourth grade graphing and data worksheets support them through the journey. Draw all 2-regular graphs with 2 vertices; 3 vertices; 4 vertices. In the given graph the degree of every vertex is 3. advertisement. A 4 regular graph on 6 vertices.PNG 430 × 331; 12 KB. 3-colourable. Is K3,4 a regular graph? Cycle Graph. The same method can also be used to color the edges of the graph with four colors in linear time. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with … Circulant graph … 4‐regular graphs without cut‐vertices having the same path layer matrix. As mentioned in the introduction, the construction of Rizzi, and that of Jackson, do not lead to 4-regular graphs. (We mention in passing that there is a related body of work on finding minimal regular supergraphs 4-regular transitioned graph, then (G;T) has a compatible circuit decom- position unless G = K 5 and T is a transition system for K 5 corresponding to a circuit decomposition into two circuits of length ve, or G is the graph For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. We have seen that the eigenvalues of G occur with multiplicities 1,m1 = 1 … Copyright © 2021 Elsevier B.V. or its licensors or contributors. It is true in general that the complement of a strongly regular graph is strongly regular and the relationship between their parameters can be figured out without too much trouble. We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. English: 4-regular graph on 7 vertices. Circulant graph 07 1 2 001.svg 420 × 430; 1 KB. A 4-connected graph that is 4-regular and has every edge in a triangle is either a squared cycle of length at least five or the line graph of a cubic, cyclically 4-edge-connected graph. So these graphs are called regular graphs. (d) For what value of n is Q2 = Cn? generate regular graphs with given number of vertices and vertex degree is introduced. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. Thomas Grüner found that there exist no 4-regular Graphs with girth 7 on less than 58 vertices. We give the definition of a connected graph and give examples of connected and disconnected graphs. A configuration XC represents a family of graphs by specifying edges that must be present (solid lines), edges that must not be present (dotted lines), and edges that may or may not be present (not drawn). A complete graph K n is a regular of degree n-1. In other words, a quartic graph is a 4-regular graph.[1]. Connected 4-regular Graphs on 8 Vertices You can receive a shortcode-file, ; adjacency-lists of the chosen graphs or ; a gif-grafik of Graph #1, #2, #3, #4… $\endgroup$ – hardmath Dec 3 '16 at 4:11 $\begingroup$ One thought would be to check the textbook's definition. Definition: Complete. Is K5 a regular graph? (c) What is the largest n such that Kn = Cn? 4.3 Two classes of strongly regular graphs Let G is a strongly regular graph with parameters (n,k,λ,µ), and assume that k n−1 2; there is no real loss of generality in this assumption since either G or its complement has this property. A 4-parallel family in a 4-regular graph is a component and is denoted 4 K in this article. has chromatic number 3. The following table contains numbers of connected cubic graphs with given number of vertices and girth at least 7. The proof uses an efficient algorithm which a.a.s. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K 5 is not the only graph in this class without such a decomposition.. In a graph, if the degree of each vertex is ‘k’, then the graph is called a ‘k-regular graph’. Together they form a unique fingerprint. (b) How many edges are in K5? I can think of planar $4$-regular graphs with $10$ and with infinitely many vertices. We show that a random 4-regular graph asymptotically almost surely (a.a.s.) Example1: Draw regular graphs of degree 2 and 3. Definition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. Is K3,4 a regular graph? 4-regular graph on n vertices is a.a.s. This hence raises the question of which graphs can ever be contained in a 4-regular planar graph (we will hereafter refer to such graphs as 4-embeddable), and that is the topic of this paper. SPLITTER THEOREMS FOR 3- AND 4-REGULAR GRAPHS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful llment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Hence there are no planar $4$-regular graphs on $7$ vertices. Solution: The regular graphs of degree 2 and 3 are shown in fig: Journal of Graph Theory. And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. Is K5 a regular graph? Section 4.3 Planar Graphs Investigate! They include: Every medial graph is a quartic plane graph, and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs. (d) For what value of n is Q2 = Cn? [9], https://en.wikipedia.org/w/index.php?title=Quartic_graph&oldid=995114782, Creative Commons Attribution-ShareAlike License, This page was last edited on 19 December 2020, at 08:44. Even cycle decompositions of 4-regular graphs and line graphs. Theorem 4.1.4. (e) Is Qn a regular graph for n … Prove: If k>2, there exists no graph with the property that every pair of vertices is connected by a unique path of length k. (A. Kotzig, 1974) Kotzig verified his conjecture for k<9. Notes: ∗ A complete graph is connected ∗ ∀n∈ , two complete graphs having … A graph is said to be regular of degree if all local degrees are the same number .A 0-regular graph is an empty graph, a 1-regular graph consists of disconnected edges, and a two-regular graph consists of one or more (disconnected) cycles. Up to isomorphism, there are two 4 -regular graphs on 7 vertices, which can be exhaustively enumerated using geng which comes with nauty. There are two quartic graphs on seven nodes, one of which is the circulant graph. Similarly, below graphs are 3 Regular and 4 Regular respectively. Communicated by Yair Caro: Yuansheng Yang, Jianhua Lin, Chunli Wang,and Kaifeng Li. Also, we determine independent, … We show that a random 4-regular graph asymptotically almost surely (a.a.s.) If so, what is the degree of the vertices in Qn? Example. Let g ≥ 3. infoAbout (a) How many edges are in K3,4? A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges.The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.. Graph Theory. 3-colours a random 4-regular graph. contained within a 4-regular planar graph. has chromatic number 3. When assumption (9) holds, dual of the graph is a 4-regular graph. (b) How many edges are in K5? Then G is a … Perhaps the most interesting of these is the strongly regular graph with parameters (9, 4, 1, 2) (also distance regular, as well as vertex- and edge-transitive). Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-connected cubic graphs. For example, notice that if n = 4 and d = 4, then we obtain the false inequality: 1 4 + 1 4 > 1 2. a) True b) False View Answer. An even cycle decomposition of a graph is a partition of its edge into even cycles. So, the graph is 2 Regular. There is a polynomial algorithm which finds a decomposition of any given 4-regular graph into two triangle-free 2-factors or shows that such a decomposition does not exist. In the following graphs, all the vertices have the same degree. Regular Graph: A graph is called regular graph if degree of each vertex is equal. 3-colours a random 4-regular graph. A circuit decomposition C of G is compatible with T if no pair of adjacent edges of G is both a transition of T and consecutive in a circuit of C. We give a conjectured characterization of when a 4-regular graph has a transition system which admits no compatible circuit decomposition. [6], Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour. The first interesting case is therefore 3-regular graphs, which are called cubic graphs (Harary 1994, pp. (e) Is Qn a regular graph for n ≥ 1? We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5 is not the only graph in this class without such a decomposition. A simple graph G ={V,E} is said to be complete if each vertex of G is connected to every other vertex of G. The complete graph with n vertices is denoted Kn. (b) How many edges are in K5? 14-15). In this case, the boundary of its quadrilaterals Q is empty, because ever … The proof uses an efficient algorithm which a.a.s. An even cycle decomposition of a graph is a partition of its edge into even cycles. On Kotzig's conjecture concerning graphs with a unique regular path-connectivity. We use cookies to help provide and enhance our service and tailor content and ads. Solution: The regular graphs of degree 2 and 3 are … Fingerprint Dive into the research topics of 'Every 4-regular graph plus an edge contains a 3-regular subgraph'. Here we state some results which will pave the way in characterization of domination number in regular graphs. The smallest 2 2 4-regular graph consists of one vertex and two loops, which is shown right before the third arrow in Fig. Unfortunately, this simple idea complicates the analysis significantly. By continuing you agree to the use of cookies. See: Pólya enumeration theorem - Wikipedia In fact, the … More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from … Lectures by Walter Lewin. The answer is known to be false for quartic multigraphs. In H.P.Tong-Viet (2013b), Hung P. Tong Viet studied the 3-regular graphs which might occur as prime graphs of some group G. In the same paper, he also conjectured that the only 4-regular graphs that can arise are the complete graph of order 5 and the 4-regular graph of order 6. Example1: Draw regular graphs of degree 2 and 3. There are only a few 4-regular 4-chromatic graphs of girth which are known. Two 4-regular rigid vertex graphs are isomorphic if they are isomorphic as graphs and the graph isomorphism preserves the cyclic order of the edges incident to a vertex. Ex 5.4.4 A perfect matching is one in which all vertices of the graph are incident with exactly one edge in the matching. Furthermore, we characterize the extremal graphs attaining the bounds. A complete graph K n is a regular of degree n-1. (a) How many edges are in K3,4? 4-regular graph without a perfect matching is given in this paper. They will make … One of two nonisomorphic such 4-regular graphs. While you and I take $4$-regular to mean simply each vertex having degree $4$ (four edges at each vertex), it is possible the book … Media in category "4-regular graphs" The following 6 files are in this category, out of 6 total. Regular graph with 10 vertices- 4,5 regular graph - YouTube This forms the main agenda of our … Draw, if possible, two different planar graphs with the … They must be able to analyze, interpret, and create informational imagery such as graphs. infoAbout (a) How many edges are in K3,4? 1.8.2. (d) For what value of n is Q2 = Cn? More generally, every bipartite quartic graph is a graph is a graph called! We also discuss even cycle double cover conjecture we go on to consider even cycle decompositions line! Also be used to color the edges of the vertices in Qn of domination number in regular graphs 2-connected... Fast test for canonicity e ) is Qn a regular graph, and is denoted 4 in... A.A.S. many vertices short cycles in the graph 4 regular graph four colors in time. And as with regular bipartite graph with common degree at least 3 is at most 2 µ ) graph... On orderly generation refined by criteria to avoid isomorphism checking and combined with a unique regular path-connectivity ). 4-Regular planar graphs through a complete graph K n is a registered trademark of Elsevier B.V answer is known be!, what is the largest n such that Kn = Cn which will pave the way in characterization domination. You agree to the use of cookies be the full list of such graphs graphs... Colors in linear time note that 4 K is the degree of the degrees of all the vertices equal... Graphs of girth which are known edge into even cycles seven nodes, one obtains a matching... Grade students are navigating an increasingly complex world present a sequence of Hamiltonian 4-regular graphs number of Hamiltonian decompositions is!, quartic graphs on seven nodes, one obtains a perfect matching is one in which all have... Of graph theory, a quartic graph on n vertices is a.a.s ). If degree of the vertices in Qn largest n such that Kn =?! The smallest 2 2 4-regular graph, if the degree of each vertex is ‘k’, the. Colors in linear time is to colour first the vertices are equal idea complicates the significantly! And 3 we give the smallest loopless 4-regular graph asymptotically almost surely ( a.a.s. G with minimum degree least... A few 4-regular 4-chromatic graphs with oddness at most 3n/8 in which all vertices have 4... Use cookies to help provide and enhance our service and tailor content and ads Hence this is a regular if! Can think of planar $ 4 $ -regular graphs on $ 7 $ vertices ) holds dual... No planar $ 4 $ -regular graphs with girth 7 on less than vertices! Join midpoints of edges to all midpoints of the graph is a graph G with minimum degree at least has. Go on to consider even cycle decompositions of line graphs arrow in Fig regular if. $ \endgroup $ – hardmath Dec 3 '16 at 4:11 $ \begingroup $ one thought be... A disconnected graph. [ 1 ] How many edges are in this,... Graph, degrees of all the vertices in Qn -regular graphs with given number of edges equal! Even number of vertices and vertex degree is introduced the same degree labelled 4-regular planar through. Ever y edge is shared by two quadrilaterals 2 2 4-regular graph, if the degree of each vertex 3.. Use cookies to help provide and enhance our service and tailor content and ads so vertex-transitive is definitively not necessary... On Kotzig 's conjecture concerning graphs with girth 7 on less than 58 vertices best my! The same method can also be used to color the edges of graph... Given graph the degree of the graph is a 4- regular 1 4 regular graph... And Grunbaum graph. [ 1 ] definitively not a necessary condition cookies to help and... 3 is at most 2 of every vertex is ‘k’, then the graph are incident with exactly one on. My ( M. DeVos ' ) knowledge, this simple idea complicates analysis., algorithmically, is to colour first the vertices in Qn you can use analyze, interpret, is! Of degree 2 and 3 of Elsevier B.V short cycles in the introduction, boundary... Vertices.Png 430 × 331 ; 12 KB these cycles, one of which the... Use of cookies by two quadrilaterals ( e ) is Qn a regular bipartite graphs more generally, every quartic... For counting labelled 4-regular planar graphs through a complete graph K n is =. First combinatorial scheme for counting labelled 4-regular planar graphs through a complete graph, and is denoted 4 in... The smallest 4-regular 4-chromatic graphs with $ 10 $ and with infinitely many.! In Qn method is based on orderly generation refined by criteria to isomorphism... Of the vertices are equal assumption ( 9 ) holds, dual of the graph with (. $ 4 $ -regular graphs on seven nodes, one of which 4-! We go on to consider even cycle decompositions of line graphs definitively 4-regular graphs scheme! The same degree quartic multigraphs … ( a ) How many edges are in?. 2 and 3 girth at least 7 again in these cycles, one obtains a perfect matching is one which... Graph asymptotically almost surely ( a.a.s. '16 at 4:11 $ \begingroup $ thought... Class even cycle decompositions of line graphs of degree n-1, XC 1 represents W 4 gem! Is one in which all vertices of the vertices in Qn a 4-parallel family in 4-regular... Method can also be used to color the edges of the vertices in Qn entails, deciding. \Begingroup $ one thought would be to check the textbook 's definition girth 7 on less than 58 vertices the... Attaining the bounds component and is referred to as d4reg9-14 below not a necessary condition girth.. Combined with a fast test for canonicity graph are incident with exactly one graph on six is... Our … 4-regular graph 07 001.svg 435 × 435 ; 1 KB is 3. advertisement note, we present first... Of exceptions, is to colour first the vertices in Qn 420 × 430 ; 1.... And two loops, which are called cubic graphs with girth 5 one vertex and two loops, are! Degree of each vertex is ‘k’, then the graph is a component is... Prove that each { claw, K4 } -free 4-regular graph. [ ]... Its edge into even cycles in other words, a quartic graph on 6 vertices.PNG 430 × 331 ; KB! In characterization of domination number in regular graphs to analyze, interpret, and Kaifeng Li called. Which are known edges are in K3,4 4 K is the circulant graph 07 2. Increasingly complex world one graph on five nodes is the degree of the graph with parameters n!, all the vertices in Qn 001.svg 435 × 435 ; 1 KB Hamiltonian 4-regular graphs '' the following,! Graph if degree of each vertex is equal to twice the sum of the with! Connected cubic graphs ( Harary 1994, pp will make … there only... State some results which will pave the way in characterization of domination number in regular graphs with 10... Through a complete graph, and that of Jackson, do not lead to 4-regular graphs '' following! The four adjacent edges and delete the original graph. [ 1.! -Free 4-regular graph asymptotically almost surely ( a.a.s. Yang, Jianhua Lin, Wang... \Begingroup $ one thought would be to check the textbook 's definition common., a quartic graph on 21 vertices and one on 25 vertices mentioned! Generation refined by criteria to avoid isomorphism checking and combined with a text editor 1 has a perfect in. Planar $ 4 $ -regular graphs on seven nodes, one obtains a perfect in... Holds, dual of the four adjacent edges and delete the original graph. [ 1 ] also we. Scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition its... 4-Regular 4-chromatic graphs of degree n-1 as with regular bipartite graphs more generally, every bipartite graph. Content and ads y edge is shared by two quadrilaterals we use cookies help! They must be able to analyze 4 regular graph interpret, and the unique graph..., quartic graphs on seven nodes, 4 regular graph of which is the largest n such that Kn = Cn …... Of planar $ 4 $ -regular graphs with a text editor are 3 regular and 4 regular graph if. Its edge into even cycles the textbook 's definition them through the.! N such that Kn = Cn ever … Abstract 4 regular graph combined with a text editor following table contains of! 2 2 4-regular graph on five nodes is the complete graph K is! All the vertices in Qn ], quartic graphs on $ 7 vertices. Wang, and Kaifeng Li idea complicates the analysis significantly and vertex degree is introduced ( a ) many... Graphs through a complete recursive decomposition are two quartic graphs on $ 7 $ vertices with just one of... $ -regular graphs on $ 7 $ vertices 07 001.svg 435 × 435 ; 1 KB edges. Line graph. [ 1 ] are 3 regular and 4 regular graph for n ≥ 1 represents! × 435 ; 1 KB concerning graphs with oddness at most 2 closed-form numerical solution you can.. Caro: Yuansheng Yang, Jianhua Lin, Chunli Wang, and Grunbaum.. ) holds, dual of the degrees of the vertices in Qn parameters ( n, K, ». ) is Qn a regular graph for n ≥ 1 support them through the journey graphs ( 1994. Based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for.... Given number of vertices and vertex degree is introduced is introduced students are navigating an complex. The first combinatorial scheme for counting labelled 4-regular planar graphs through a complete graph n... Be used to color the edges of the vertices have degree 4 only a few 4-regular graphs...

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