Fronček, Dalibor

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

Commentationes Mathematicae UniversitatisCarolinaeDalibor FrončekOn graphs with prescribed edge neighbourhoodsCommentationes Mathematicae Universitatis Carolinae, Vol. 30 (1989), No. 4,749--754Persistent URL: http://dml.cz/dmlcz/106797Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1989Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copyof any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronicdelivery and stamped with digital signature within theproject DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czComment.Math.Univ.Carolinae 30,4(1989)749-754 749 On graphs with prescribed edge neighbourhoods DALIBOR FRONCEK Abstract. Let G be a graph and let / be its edge. Then /V£(/) is the subgraph of G induced by the set of ai! vertices adjacent to at least one of the end vertices of / . In the paper some classes of graphs with the prescribed properties of iV£(/) are studied. Keywords: Local properties, neighbourhood of an edge, neighbourhood of a vertex Classification: 05C99 1. Introduction. Afl graphs considered in this article are finite undirected graphs without loops and multiple edges. Let G be a graph, let / be its edge with end vertices x,y. By the symbol Nfa(f) or N@(xy) we denote the subgraph of G induced by the set of all vertices of G which are not incident to / and are adjacent to at least one end vertex of / . The graph No(/) W*H °e called the edge-neighbourhood (or e-neighbourhood) of / in G. By the symbol N%(f) or N^(xy) we denote a closed e-neighhourhood, i.e. the subgraph of G induced by the vertices x, y and the set of vertices of G which are adjacent to at least one vertex of the pair x,y. A graph induced by the vertex se M is denoted (M). Zeiinka [6] proposed the e-neighbourhood version of the well-known Zykov's prob-lem [7] (concerning vertex-neighbourhoods) : Characterize the graphs H with the property that there exists a graph G such that N&(f) S H for each edge / of G. The graph If with the above mentioned property will be called an e-realizable graph and G will be called the e-realization of E. The se of afl e-realizations of B wifl be denoted $e(JEf), the class of afl e-realizable graphs will be denoted Ht* Let P be some prescribed property of graphs. If the e-neighbourhood of each edge f of G has the property P then G will be called a graph with prescribed e-neighbourhood. Zeiinka [6], Nedela [5] and the author [2], [3], [4] studied some classes of e-realizable graphs and also showed some graphs which are not e-realizable. hi this article we study some properties of graphs with prescribed e-neighbour-hoods and show some e-realizable and non-e-realizable graphs. 2. Some properties of graphs with prescribed e-neighbourhoods. At first we shall prove some simple lemmas. 750 D.FVonček Lemma 1, Let G have no triangles. Then both NG(f) and NG(f) are bipartite graphs for each edge f of G. PROOF : Suppose that NG(y\y2) of an edge f = y\y2 is not a bipartite graph. Then NG(y\y2) contains an odd cycle C2n+i = (xi, X2,..., x2n+i) and there exists a pair of adjacent vertices x, and x;+i which are both adjacent either to y\ or to y2. Thus G contains a triangle, which is a contradiction. Hence, NG(f) of each edge / of G is a bipartite graph. Now suppose that the closed ^-neighbourhood NG(yiy2) of an edge y\y2 is not a bipartite graph. As G contains no triangles, no vertex of NG(y\y2) is adjacent to both y\ and y2. Let NSCyiya) have the parts B\ =* { x i , x 2 , . . . , x r } and B2 = {z\,z2y...,za). Then at least oae of the vertices y\, y2 is adjacent to the vertices x» and Zj belonging to this same component of NG(y\y2). Without loss of generality we can suppose that y\ is adjacent to xi and simultaneously to z;, which are the end vertices of the path P2j -= (x\,Z\,x2,... ,Xj,Zj). As G contains no triangles, and xi is adjacent to y\, then z\ has to be adjacent to y2 and X2 again to y\. Analogously for each *» 1 is *' is i> if xi 1S adjacent to y\ then z, is adjacent to y2. But Zj is adjacent to y\, which is a contradiction and then neither y\ nor y2 can be adjacent to vertices belonging to both parts of any component of NG(y\y2). Thus NG(y\y2) is a bipartite graph. • If x is a vertex of G, then by vertex-neighbourhood (or u-neighbourhood) of x in G we mean the subgraph of G induced by the set of all vertices of G which are adjacent to x and we denote it by NG(x). P r o p o s i t i o n 1. Let G be an e-realization of some tree T. Let there exist a vertex xofG such that N%(x) contains a cycle Crf r > 4. Then NG(x) •=* Cr. PROOF : Suppose that NG(x) contains a cycle Cr with vertices yi,y2, • • • ,yr and the chord y\yi, 3 < i < r - 1. Then NG(xyr) contains C, with the vertex set {yi,y2» • • • i Vi] which is a contradiction. We can also see that NG(x) contains no other vertex y r +i. In the opposite case NG(xyr+\) contains Cr with the vertices yi,y2, • *.,yr, which is again a contradic­tion. • Lemma 2. Let G be an e-realization of a path Pn, n > 4. Let an edge y\y2 belong to at least two triangles. Then at least one of the graphs NG(y\)} NG(y2) is isomorphic toCri r < n + l. PROOF : As G is an e-realization of P„, NG(yxy2) is isomorphic to Pn = ( x i , x 2 , . . . , x r t ) . Let yij/2 belong to the triangles ( y i ^ x , ) and (yi,y2,x*), 1 < j<k. If k ss j -f 1, then either NG(y\Xi) (for any t / j , j + 1) contains the triangle (y2»*i>*i+i) or NG(y2X\) contains the triangle (yi,Xj,xJ +i), which is a contradic­tion. If k = j ~r 2, then the vertex x>+i is adjacent to at least one of the vertices yi, y2-Let it by yi. Then NG(x;y2) contains the triangle (yi,xJ. fi,a:J +2)i which is again a contradiction. On graphs with prescribed edge neighbourhoods 751 Now consider the case k > j + 2. Suppose that there exists an edge x,x,+i with the property that j < i < k — 1 and the vertex x, is adjacent to yi and Xi+i is adjacent to y2- Then NG(xiy\) contains a star with the center y2 and the terminal vertices Xj,Xi+i,Xfc, which is a contradiction. If Xi is adjacent to y2 and x,*+i is adjacent to yi, we get a contradiction in the same way. Thus all the vertices Xj+\,...,Xk-\ are adjacent to the only vertex of the pair yi,y2- Without losing generality we can suppose that it is yi- Then NG(y\) contains Ck-j+2 with the vertices y2, £ j , Xj+\,..., x* and according to Proposition 1 NG(y\) is isomorphic to Ck-j+2- The inequality k — j + 2 = r ^ n - r l obviously holds - in the opposite case NG(y\y2) contains at least n + 1 vertices. • L e m m a 3 . Let G be an e-realization of a path Pn, n > 4. Then G has no triangles. P R O O F : Suppose that G contains a triangle (xo,xi,X2). Then N£(x0xi) contains some vertex X3 which is adjacent to x2 and simultaneously to at least one of the vertices xo, xi. Let it be x0 . Thus the edge x0X2 belongs to two triangles, and according to the assertion of Lemma 2 the v-neighbourhood of at least one,of the vertices xo,X2 is isomorphic to a cycle. Without los of generality we can suppose that NG(x0)^Cr = ( x i , x 2 , . . . , x r ) . Now we shall distinguish two cases. We begin with the simpler case r = n + l . As G G 7le(Pn), then NG(x\X2) = Pn and thus at least one of the vertices X3, xn+\ = x r is adjacent to some vertex xn+2- Let it be xn+i . Hence N£(xoxn+i) contains n + 1 vertices x\,..., xn , xn+2, which is a contradiction. Now let us investigate the case r < n -f 1. As we supposed that NG(XQ) = Cr = (xi,X2,. . . , x r ) , NQ(XQXT) must contain a vertex z\ which is adjacent to at least one of the vertices Xi ,x r_i . Without loss of generality we can suppose that it is x r _i . As z\ is from N£(x0xr), it has to be adjacent to x r (if it is adjacent to xo, then NG(xoZ\) contains the cycle (xi ,X2, . . . ,x r ) ) . Thus the edge x r _ix r belongs to two triangles and the ^-neighbourhood of at least one if its end vertices is isomorphic to Ca. Let NG(xr) = Ca. This cycle is induced by the vertices x i , x o , x r _ i , z i , . . . ,2,-3 because the cycle (xi,X2,.. . , x r ) has no chord. But hence NG(x0xr) =* Cr+,-.4 = ( « ! , . . . , x r_i , zu • • • > 2*-3>» which is again a contradiction. • Lemma 4. Let a connected graph G belonging to 7le(H) for a graph H be not regular. Then G is bipartite if and only if it has no triangles. PROOF : ( = > ) is trivial. ( < = ) Let G have no triangles. Then the equality (1) d e g x + d e g y = k holds for each pair of adjacent vertices x, y and k is the* number of vertices of N&xy) S* H. As we suppose that G is not regular, there exists a pair of adjacent vertices x©, x\, such that degxo ^ degxi . It follows from (1) that degx^ = degxi for each vertex Xi adjacent to x0 and, analogously, degxy = degx0 for each vertex Xj adjacent 752 D.Fron£ek to x\. Because G is connected we can easily see that each vertex of G is either of degree degx0 or of degree degxi and no vertices with the same degree can be adjacent to each other. Thus G is a bipartite graph. • Corollary 1. Let a connected e-realizable graph H have an odd number of vertices, let G € *R,e(H). Then G is a bipartite graph if and only if it has no triangles. PROOF : ( = > ) is trivial. ( < = ) If G has no triangles and H has an odd number of vertices, G is not regular and our assertion follows from Lemma 4. • Now we shall present further simple propositions. Proposit ion 2. Let a connected e-realizable graph H have an odd number of ver-tices, let G € %€(H) be regular. Then G contains a triangle. PROOF : Let No(yiy2) = (*i, #2, • • •, £ 2 n + i ) . Then somevertex x, has to be adja-cent to both yi and y2 and G contains the triangle {yi,y2,-c.) and also (yi,y2,-C;) for a neighbour Xj of a,j. • Proofs of the following trivial propositions are left to the reader. Proposition 3 . Let a regular graph G be an e-realization of H. Let t(e) be the number of triangles which contain an edge e. Then t(e) =- t(f) for any edges e, / of G and if H is connected, then t(e) > 2. Proposition 4. Let H be a hamiltonian graph and let G € TZe(H) have no trian-gles. Then H is bipartite with parts Pi. P2; IP^ = |P2 | = k and G is regular of degree k + 1. Proposition 5 . The graph nK2 is e-realizable by Qn+i-3 . Which paths are e-realizable ? Now we turn our attention to the e-realizability of paths, from Lemma 3, Corol-lary 1 and simple considerations concerning the number of vertices of Pn we can easily see that the following Theorem holds. Theorem 1. Let G 6 Ht(Pn)f n £ 4. Then (t) if n ~~2k, G ha3 no triangles and it is regular of degree k + 1 ; (tt) ifn SB 2k + 1, G is bipartite and bi regular of degrees k + 1 and k + 2. Brown and Connelly [1] proved that all paths with the simple exception P3 are v-reaHzable and on the other hand Zelinka [6] showed that P2 and P3 are e-realizable. We shall solve the above mentioned problem for n -= 4 ,5,6 in the next theorem. Theorem 2. The paths P4, P5 and P6 are not e-realizable. PROOF : Suppose that G € TMPi). Then the e-neighbourhood of an edge yiy2 is isomorphic to P* -* (x\,..., £4) andwithout losing generality we can suppose that y% is adjacent just to the vertices £1,23, while y2 is adjacent just to xt^x*. Now investigate NQ(X$XA). AS it contains P3 =- (yi,y2,ar2), there exists a vertex z of On graphs with prescribed edge neighbourhoods 753 G such that it is adjacent to x2 (z cannot be adjacent to y% because it does not belong to NG(yiy2)) and simultaneously to x* (z cannot be adjacent to 3:3 because in this case G contains the triangle (x2,x$,z)). But in this case degX2 s£ 4, which is a contradiction to the assertion (i) of Theorem 1. Thus P4 is not e-realizable. Now suppose that G € 7^e(Ps)- Then NG(y\y2) is isomorphic to P5 = (xi, x2,..., X5) and without loss of generality we can suppose that y\ is adja-cent just to x\,x$,xs and y2 just to #2,3:4. Note that y2,x\,x$,x$ are of degree 3 and yi, x2, £4 are of degree 4. Now we shall investigate NG(y\ £i) which contains the vertices y2,^2»^3,£5 and some other vertex z. Because G € %€(P&), z is adjacent to x\ and thus N£(yi£i) 3 P5 = (y2,£2,£3,£5,*)- As (y2,£2,£3) — P3 and x5 is adjacent neither to y2 nor to £3, z has to be adjacent to £5 and to just one of the vertices £3,y2- But z cannot be adjacent to £3 (in the opposite case deg£3 *> 4) and as z does not belong to NG(y!y2), it also cannot be adjacent to y2. Hence P5 is not e-realizable. At last suppose that G is an e-realization of P6 and NG(y\y2) = P6 = ( x i , £ 2 , . . . ,£e)- Again without losing generality let y\ be adjacent just to £i>£3*£s and y2 to X2,x4 ,x6 . NG(yix3) contains P5 = (xx,x2,y2,£4,x5) and thus G has to contain a vertex z adjacent to £3 and to exactly one of the vertices £1, £5 which are terminal of the path P5 mentioned above. If z is adjacent to £5, then NG(xsyi) contains P5 = (z, £3, £4, y2, £6) and the ver-tex £1 . Because £1 cannot be adjacent to £6 (in the opposite case NG(yiy2) -= Ce) it has to be adjacent to z. But hence NG(x$yi) contains C$ with the vertices £i,£2,y2,£4,£5,z, which is a contradiction and thus z is not adjacent to X5. Therefore we shall suppose that z is adjacent to x i . Then NG(y\X\) contains P4 = (z,xz,x2,y2) and the vertex X5. As y2 is of degree 4, according to Theorem 1 it cannot be adjacent to any other vertex of G and thus y2 is the terminal vertex of P6 = NG(y\Xi). Hence z has to be adjacent either to X5 (but in this case NG(yiXs) contains Ce with the vertices £s,z,£i ,£2,y2,£4), or to another vertex v of NG(y\xi). Because v does not belong to NG(y\y2) it has to be adjacent to Xi. But we supposed that z is also adjacent to x\ and then the vertices X\,z,v induce the triangle. Therefore, according to the assertion of Lemma 3, G is not an e-realization of Pe. As for n > 7 no e-realizability of P„ is known and we are not able to prove its non-e-realizability by the methods used above, we propose our problem. Problem. Which paths Pn for n > 7 are e-realizable? • REFERENCES [1] M.Brown, R.Connelly, On graphs with a constant link I and II,, Proof Techniques in Graph Theory (F.Harary ed.)t Academic Press (1969) and Dicrete Math. 11 (1975), 199-232. [2] D.Froncek, Graphs with given edge neighbourhoods, Czech. Math. Journ. 39 (1989), 627-630. [3] D.Froncek, Edge-realizable graphs with universal vertices, submitted. [4] D.Froncek, Graphs with near v- and e-neighbourhoods, Glasgow Math. Journ., to appear. [5] R.Nedela, Which cycles are edge-realizable ?, Czech. Math. Journ., submitted. [6] B.Zeiinka, Edge neighbourhood graphs, Czech. Math.Journ. 36 (1986), 44-47. 754 D.Fronček [7] A.A.Zykov, Problem 30, Theory of graphs and its applications Proč. Symp. Smolenice 1963, Academia Prague (1964), 164-165. Katedra matematiky a DG VŠB, Vítězného února, 708 33 Ostrava, Czechoslovakia (Received March 28,1989) 

On graphs with prescribed edge neighbourhoods

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On graphs with prescribed edge neighbourhoods

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