Kotzig, Anton

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On Even Regular Graphs of the Third Degree

Institute of Mathematics AS CR, v. v. i.

Matematicko-fyzikálny časopisAnton KotzigOn Even Regular Graphs of the Third DegreeMatematicko-fyzikálny časopis, Vol. 16 (1966), No. 1, 72--75Persistent URL: http://dml.cz/dmlcz/126715Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1966Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czM A T K M A T í C K O - F Y Z I K A L N Y OASOIM* SAV. Iť ON EVEN REGULAR GRAPHS OF THE THIRD DEGREE A N T O N KOT/ZUJ. Brat is lava 111 the present paper we mean by ..graplF' a finite, non-oriented graph with out loops. Let 6/be any even regular graph of the third degree without multiple edge-It is known that the number of vertices of a regular graph of the third (h^gif̂  is always even (see, e. g. Konig, [3|, p. 21, theorem 3). Let 2m be the number of vertices of G. As G is an even graph and without nmJtipJe cd^cs, \t neeossarih follows that 2m ^ 0. Let G' or G" respectively be the graph arising from <7 by the splitting of its edge h that joins the vertices \i, v (see Fig. 1 the eon cept of the splitting of edges was originated by Prink [2]). Fig. 1. N o t e 1. An even graph is also called bichromatic graph (see Berge, [1 | , p. 30). The name was adopted because of the fact that the vertices of such a graph can always be coloured by two colours in such a way that any edge joins two vertices of different colours. In the figures we shall make use of this possibility so that the vertices of one colour will be marked by full circles, the others by void circles. 72 The following is evident: Any edge of an even graph without multiple edges may be split in two ways; by both ways we always get an even regular graph of the third degree. We shall say that the edge h of the graph G is X-reducible, if at least one of the graphs arising from the splitting of h does not contain, multiple edges (in the reverse case we shall say that the edge h is X-irreducible). Such a splitting of the edge h, where there arises from the graph G a graph without multiple edges, will be called the X-reduction of the graph G on the edge h. Lemma 1. Let G be an even regular graph of the third degree without multiple, edges. An if edge h of it is X-irreducible if and only if it belongs to at least two different quadrangles of G that have — apart from h -- ett least another edge in conunon. This common edge is always adjacent to h. Proof. Let h be any edge of the graph G and let h join the vertices H, v. The vertices joined in G by an edge of the vertex u, or v respectively, will be denoted as seen in Fig. 1. I t is evident that a, b, c, el are four different vertices. There may exist such quadrangles in G that by proceeding along them we pass through the vertices of the graph in the following order (see Fig. 1): quadrangle Q\. u. v. c. a: quadrangle Q-z- w, ?\ c, b\ quandrangle Qi: u, v, tf, a: quadrangle Q\: u. v. d, b. If there existed in G from among the four considered quadrangles only the quadrangles Qi, Q\ that -- apart from h have no other edge in com-mon (or if only one of them existed) — then the graph G" would not contain a multiple edge and the edge h would be X-reducible. Similarly, if in G there existed only the quadrangles Qz^Qz (from among the four considered quadran gles). then the graph G' would not contain, multiple edges and the edge h would be A"-reducible. If both the graph G' and the graph G" are to contain multiple edges, there must exist in G at least one of the following four pairs of quadran gles: {Qi.Q-i}, {Q\> Q3}, {Q2, Qi}, {$3, $4}- "Each of the mentioned four pairs of quadrangles has the following property: the quandrangles of the pair have — apart from h - another edge in common and this edge is adjacent to h. The lemma is proved. We shall sayr that en even regular graph of the third degree is X-irreducible if each of its edges is A'-irreducible. The graph in Fig. "2 is an example of such a graph. I t will be denoted further by the symbol G*. Theorem 1. Each component of an X-irre-ducible graph is isomorphic with the graph G*. Proof. I t is evident that the component of an Ar-irreducible graph is an X-irredu-7 * cible g r a p h . It sut'iioes therelbi'o to prove that each connected A'-irredu­cible graph is isomorphic to f»'*. Let (! be a connected K-irroducible graph and lot // bo any of its edges: let //. r be vert ices incident a t h. T h e edge h is A'-irreducible. According to lemma 1. t h e IHUJV h belongs to two different quadrangles Q, Q'. which, apar t from //. have a n o t h e r odg(x in common (let us d e n o t e it by j). and j . h are ad jacent . We can assume wi thout loss of general i ty that the vd'iiv <j joins t h e \'ertox /• with a certain vertex /!\ Tt is ev ident that the quadrangles (f Q' c a n n o t lia\'e three ( dges in common (as the foui'th edge from Q and t h e fourth cd^v from Q' would bo mult iple edges, which, is impossible in a N-irreducible g r a p h ) . Jt follows t h a t the graph (1 contains as a partial subraph the graph F. given in Fig. .», where r. * are two different vertices. KІL Lot us d e n o t e by the symbol fr (or/,-, or /',-. respectively) that edge incident at t h e vertex /' (or /\ or >\ respectively) t h a t docs not belong to F a n d let /"• (or )\ or s\ respectively) be t h e second vertex, a t which the edge /• ( o r / , or / s , respectively) is incident. T h e edge fr is A'-irreducible. I1 hero lore//• must belong t o a certain quadrangle Q". This q u a d r a n g l e conta ins r and m u s t there­fore conta in further e i ther the edge joining r with //. oj' t h e iK\^io joining r with w. Hence: eitlnT /"• — l\ or r *. By a similar considerat ion with respect to t h e edge /• a n d / , wo find t h a t r r =-- «s holds. But tlien t h e graph F with t h e edges/- ./- ,/• a n d with t h e vortex r r .>• forms a g raph t h a t r e p r e s e n t s the whole graph (} a n d (1 is isomorphic to (7*. This proves t h e t h e o r e m . Let (/ be a n y even regular graph of t h e t h i r d degree w i t h o u t mu l t ip le edges a n d let N j To, Vi} be such a decomposit ion of t h e set of vertices of t h e graph (} t h a t t h e following holds: a n y edge from (} joins vertices from different classes of t h e decomposit ion F. N o t e 2. I t is known t h a t if n is t h e n u m b e r of c o m p o n e n t s of t h e even g r a p h (1 and /// is t h e n u m b e r of decomposi t ions with t h e above p r o p e r t y , t h e n m 2" L hence in an even g r a p h t h e r e exists a t least one such decompos i t ion. Let u{) r r>) be vertices from V(, a n d ?/i -,- r\ vertices from V\ such t h a t 1 in the uraph (I there exis ts an edge g joining the vert ices Ho, U\ and there exists the (Mlg(4 // joining the vert ices v(). rL . If t he g r a p h U' arises from (J in the way shown in Fig. 4. we shall say tha t (V arose from O b y an .Y-exlension on the edge-- g, h. Hence A-ex tens ion is t h e inverse process of A'-reduc t ion. «o W1 *0 V l Lemma 2.: Lei (i be <iny even regular graph of the third degree without multiple edges and lei g, h be (wo of its edges that are not adjacent. At the edges g, h exactly OH( X-(xtension is possible and the graph (?' which arises in this wag is ahcay* an (ven regular graph of the third degree without multiple edges. T h e p r o o f is ev ident . N o t e 3. The A"-extension is defined only on pairs of edges t h a t are not adjacent . I t we ex tended t h e g raph (J in a similar way on two ad jacen t edges, then the g raph c/' would contain mul t ip le edges, (see .Fig. 4). Theorem 2. Any even regular graph of the third degree with 2n vertices withont multiple edges is either X-irreducible, or it may be constructed by repeated X-ex-tensions from a certain X-irreducible graph with /// components, which are all isomorphic to the graph, (/*, where tim < 2u. P r o o f . Theorem 2 is a direc t consequence of theorem 1, of the definition of the A^-irreducible graph and of t he relation between A'-reduction and A'-ex-tension. R K F K R F N F F S ! I | K r i ' u c ( \ . Tbenric des gra>pbes et ses applications* Paris 1958. |L>| F r i n k (). . A Proof of Petersen* Theorem,, Ann. Math . 27 (192(>), 491 -493. ! .'> J K d n i u i). . Tbeoi-ie der endlicben nnd nnendlieben (rraj)hen, Leipzig 19.,(>. K( (I Maivh li. 1965, Katedra nunierickej ni-atema-tiky a nbatematickej statistiky Prtrodoredeckej fakulty lT ni ret žity Komenského, Pra(islar<i 7 5 

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On Even Regular Graphs of the Third Degree

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Institute of Mathematics AS CR, v. v. i.