MINIMUM CROSSINGS IN JOIN OF GRAPHS WITH PATHS AND CYCLES

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1 3 Acta Electrotechica et Iformatica, Vol. 1, No. 3, 01, 3 37, DOI: /v MINIMUM CROSSINGS IN JOIN OF GRAPHS WITH PATHS AND CYCLES Mariá KLEŠČ, Matúš VALO Departmet of Mathematics ad Theoretical Iformatics, Faculty of Electrical Egieerig ad Iformatics, Techical Uiversity of Košice, Letá 9, Košice, Slovak Republic, maria.klesc@tuke.sk, matus.valo@tuke.sk ABSTRACT The crossig umber cr(g) of a graph G is the miimal umber of crossigs over all drawigs of G i the plae. Oly few results cocerig crossig umbers of graphs obtaied as joi product of two graphs are kow. There was collected the exact values of crossig umbers for joi of all graphs of at most four vertices ad of several graphs of order five with paths ad cycles. We exted these results by givig the crossig umbers for joi products of the special graph o six vertices with isolated vertices as well as with the path o vertices ad with the cycle o vertices. Keywords: graph, drawig, joi product, crossig umber, path, cycle 1. INTRODUCTION Let G be a graph with vertex set V (G) ad edge set E(G). A drawig of G is a represetatio of G i the plae such that its vertices are represeted by distict poits ad its edges by simple cotiuous arcs coectig the correspodig poit pairs. For simplicity, we assume that i a drawig (a) o edge passes through ay vertex other tha its ed-poits, (b) o two edges touch each other (i.e., if two edges have a commo iterior poit, the at this poit they properly cross each other), ad (c) o three edges cross at the same poit. The crossig umber cr(g) is the smallest umber of edge crossigs i ay drawig of G. It is easy to see that a drawig with miimum umber of crossigs (a optimal drawig) is always a good drawig, meaig that o edge crosses itself, o two edges cross more tha oce, ad o two edges icidet with the same vertex cross. Fidig the crossig umber of a graph is useful i may areas. The most promiet area is VLSI techology. The lower boud o the chip area is determied by crossig umber ad by umber of vertices of the graph []. It plays a importat role i various fields of discrete/computatioal geometry [19]. Crossig umber is also parameter yieldig the deviatio of the graph from beig plaar. The crossig umber iflueces sigificatly readability ad therefore it is the most importat parameter whe aesthetics of the graph is cosidered. It is used mostly i automated visualisatio of graphs. The ivestigatio o the crossig umbers of graphs is a classical ad very difficult problem. It was proved by Garey ad Johso [5] that fidig the crossig umber of a geeral graph is NP hard ad it remais NP hard eve for cubic graphs [8]. Pioeerig papers about crossig umbers were published i the secod half of the twetieth cetury. Turá formulated The brick factory problem, which arose durig World War II., by publishig i the article [0]. Give problem was a graph problem where kils ad storage yards are vertices ad the rails betwee kils ad storage yards are edges i the graph. Turá asked whether the umber of crossigs of the rails could be miimized. He realized after several days that the actual situatio could be improved, but the exact solutio of the geeral problem with m kils ad storage yards seemed to be very difficult. Usig the costructio of suitable diagram of the complete bipartite graph K m,, Zarakiewicz [] gave the upper boud of the crossig umber of the graph K m, ad cojectured that cr(k m, ) = m m 1 1. (1) The give problem has ot bee geerally solved. Kleitma [10] cofirmed Zarakiewicz s cojecture for every ad 1 m 6. Woodall proved that for 7 m 8 ad 7 10, the crossig umber of K m, is equal to Zarakiewicz umber, too [1]. The exact values of crossig umbers are kow oly for several specific families of graphs. The Cartesia product (the defiitio see i [1]) of two graphs is oe of them. Let C be the cycle of legth, P be the path of vertices, ad S be the star isomorphic to K 1,. Harary et al. [7] cojectured that the crossig umber of the Cartesia product C m C of two cycles is (m ), for all m, satisfyig 3 m. It was proved by Glebsky ad Salazar [6] that for ay fixed m, the cojecture holds for all m(m + 1). The cojecture has also bee verified for m 7. Besides of Cartesia product of two cycles, there are several other exact results. I [1, 9], the crossig umbers of G C for all graphs G of order at most four are established. I additio, the crossig umbers of G C are kow for some graphs G o five or six vertices [4, 16]. Bokal i [3] cofirmed the geeral cojecture for crossig umbers of Cartesia products of paths ad stars formulated i [9]. Crossig umbers of Cartesia products of stars ad paths with graphs of order at most five were studied i [9, 11, 1]. The table i [13] shows the summary of kow crossig umbers for Cartesia products of path, cycle ad star with coected graphs of order five. The Joi product G i + G j of graphs G i ad G j is created from vertex-disjoit copies of G i ad G j by addig all edges betwee V (G i ) ad V (G j ). For V (G i ) = m ad V (G j ) =, the edge set of the graph G i + G j is the uio of E(G i ), E(G j ) ad E(K m, ). Let us deote by D the discrete graph o vertices. The crossig umbers for joi of two paths, joi of two cycles, ad for joi of path ad cycle were studied i [14]. Moreover, the exact values for crossig umbers of G + D ad G + P for all graphs G of order at most four are give i [17]. The crossig um- ISSN (prit) c 01 FEI TUKE Uautheticated Dowload Date 1/15/14 3:08 AM ISSN (olie)

2 Acta Electrotechica et Iformatica, Vol. 1, No. 3, bers of the graphs G + P ad G +C are also kow for very few graphs G of order five ad six, see [15, 18]. Let D (D(G)) be a good drawig of the graph G. The umber edge crossigs of the graph G i D is deoted by cr D (G). For edge-disjoit subgraphs H i ad H j of G, we deote by cr D (H i,h j ) the umber of crossigs betwee the edges of H i ad the edges of H j i D ad by cr D (H i ) the umber of crossigs amog edges of H i i D. For ay three edge-disjoit subgraphs H i, H j ad H k of the graph G the followig equatios hold: cr D (H i H j ) = cr D (H i ) + cr D (H j ) + cr D (H i,h j ), cr D (H i H j,h k ) = cr D (H i,h k ) + cr D (H j,h k ). () I the proofs of the paper, we will ofte use the term regio also i oplaar drawigs. I this case, crossigs are cosidered to be vertices of the map. Let G be a graph cosistig of oe cycle abcdea, oe additioal vertex f ad two edges a f ad d f. For brevity, let C e 5 (G), C f 5 (G), ad C 4(G) deote the cycles abcdea, abcd f a ad a f dea, respectively. The aim of this paper is to give the crossig umbers of joi products of the graph G with the discrete graph D as well as with the path P ad the cycle C.. THE CROSSING NUMBER OF G + D N The graph G + D cosists of oe copy of the graph G ad vertices t 1,t,...,t, where every vertex t i, i = 1,,...,, is adjacet to six vertices of G, see Fig. 1. For i = 1,,...,, let T i deote the subgraph iduced by six edges icidet with the vertex t i ad let F i = G T i. To simplify the otatio, let G() deote the graph G + D, i this paper. I Fig. 1 oe ca easily see that G + D = G() = G K 6, = G ( T ). i (3) i=1 The graph G + D 1 cotais a subdivisio of the complete bipartite graph K 3,3, ad therefore cr(g+d 1 ) = 1. For =, cr(g+d ) sice removig ay edge of the graph G + D results i a graph cotaiig a subdivisio of K 3,3. Hece, the assertio of the Theorem is true for = 1 ad =. Suppose ow that for 3 3 cr(g( )) + (4) ad cosider such a drawig D of the graph G() that 1 cr D (G()) < 6 +. (5) Assume first that there are two differet subgraphs T i ad T j that do ot cross each other i D. Without loss of geerality, let cr D (T 1,T ) = 0. The subdrawig of D iduced by T 1 T divides the plae i such a way that there are two vertices of G o the boudary of every regio. The graph G cotais two edge disjoit stars S3 a ad Sd 3 iduced by the vertices a,b,e, f ad d,c,e, f, respectively. As each regio of D(T 1 T ) cotais exactly two vertices of G o its boudary, cr D (S3 x,t 1 T ) 1, x = a,d, ad therefore cr D (G,T 1 T ). Moreover, as cr(k 6,3 ) = 6, i D, every subgraph T k, k = 3,4,...,, crosses T 1 T at least six times. Sice G() = G+D = G( ) (T 1 T ) ad G( ) = K 6, G, usig () ad (4) we have cr D (G()) = cr D (G( )) + cr D (T 1 T ) + + cr D (K 6,,T 1 T ) + cr D (G,T 1 T ) 3 + ( ) ( ) This cotradicts (5). Thus, cr D (T i,t j ) 0 for all i, j = 1,,...,, i j. Moreover, usig () ad (3) together with cr(k 6, ) = 6 1 we have cr D (G()) = cr D (K 6, ) + cr D (G) + cr D (K 6,,G) 1 + cr D (G) + cr D (K 6,,G). This, together with the assumptio (5), implies that Fig. 1 A drawig of G + D with crossigs. Theorem.1. cr(g + D ) = for 1. Proof. The drawig i Fig. 1 shows that the iequality cr(g + D ) holds. We prove the reverse iequality by iductio o. cr D (G) + cr D (K 6,,G) < (6) ad hece, there is at least oe subgraph T i which does ot cross G i D. Moreover, it implies that G cotais at least oe iteral crossig sice G is ot outerplaar graph ad furthermore, cr D (G,K 6, ). Without loss of geerality, let cr D (G,T 1 ) = 0. There exists T i, i {,3,...,} such that cr D (G T 1,T i ) 3. (7) ISSN (prit) c 01 FEI TUKE Uautheticated Dowload Date 1/15/14 3:08 AM ISSN (olie)

3 34 Miimum Crossigs i Joi of Graphs with Paths ad Cycles If otherwise, the iequality cr D (G T 1,T i ) 4 holds for every i =,3,...,, the cr D (G()) = cr D (K 6, 1 ) + cr D (G T 1 ) + + cr D (K 6, 1,G T 1 ) ( 1) ( ) 1 1 = As , i D there are at least crossigs. This cotradicts (5). Assume ow the subdrawig D(F 1 ) of the subgraph F 1 = G T 1 iduced by D. Sice cr D (G,T 1 ) = 0, all edges of T 1 are placed i oe regio of D(G) with all six vertices of G o its boudary, say outside G. Hece, i D(F 1 ), o the boudary of every regio outside G there are exactly two vertices of G. It implies from (7) ad the fact cr D (T 1,T i ) 0 that, i D, a vertex t i, i, ca be placed oly i a regio of D(F 1 ) with at least four vertices of G o its boudary, i.e. iside G. For the case cr D (C5 e (G)) 0, iside the crossed 5-cycle C5 e (G) there are regios with at most three vertices of C5 e(g). This immediately implies that there is o regio with more tha three vertices o its boudary i D(F 1 ). The same holds if the edges of C f 5 (G) cross each other. So, cr D (C5 e(g)) = 0 ad cr D(C f 5 (G)) = 0 ad the ecessary iteral crossig i G is forced amog the edges of C 4 (G). There are oly two possible drawigs of C 4 (G) with oe crossig yieldig oly two possible drawigs of G: the edge ae is crossed by the edge d f or the edge de is crossed by the edge a f, see Fig.. Fig. Two possible drawigs of F 1 cotaiig a regio with four vertices of the graph G. Cosider ow the subdrawig of G T 1 T i iduced by D ad let us cout the ecessary crossigs betwee the edges of G T 1 ad the edges of T i. Recall that t i caot be placed outside the graph G ad hece t i ca be placed oly i oe of three possible regios. For the case of placig the vertex t i i the regio with four vertices of G o its boudary, cr D (G T 1,T i ) = 3 with cr D (G,T i ) =, or cr D (G T 1,T i ) = 4 with cr D (G,T i ) =, or cr D (G T 1,T i ) 5. I the other cases, t i is placed i a regio with two vertices o its boudary (there are oly two such regios). The T i crosses the boudary of the regio at least four times ad therefore, cr(g,t i ) 4. Hece, cr D (G T 1,T i ) 5, because cr D (T 1,T i ) 0. Assume that r is the umber of T i with cr D (G T 1,T i ) = 3 ad cr D (G,T i ) = ad that s is the umber of T i with cr D (G T 1,T i ) = 4 ad cr D (G,T i ) =. The followig iequality immediately follows: cr D (G()) = cr D (K 6, 1 ) + cr D (G T 1 ) + + cr D (K 6, 1,G T 1 ) r + 4s + 5( r s 1) 1 = 6 + ( ) r 4 s 6. This together with (5) implies 1 4 r s 4 6 < 0 ad hece, r + s > + for eve ad r + s > 1 for odd. O the other had, G cotais at most 1 crossigs ad the iequality r + s < 1 must hold. This cotradictio completes the proof. 3. THE CROSSING NUMBER OF G + P N The graph G+P cotais G+D as a subgraph. Let us deote the path t 1 t...t by P. So, it is easy to see that ( G + P = G K 6, P = G P T ). i (8) i=1 For = 1, the graph G+P 1 is isomorphic to the graph G+ 1 ad therefore, cr(g + P 1 ) = 1. Theorem 3.1. cr(g + P ) = for. Proof. The graph G+P ca be created from G+D show i Fig. 1 by addig the edges of P i such way that oly the edge bc of G is crossed by the edge t t +1 of P. This implies that cr(g + P ) O the other had, G + P cotais G + D as a subgraph ad hece, by Theorem.1, cr(g + P ) 1 +. Suppose that there is a drawig D of the graph G + P with crossigs. The o edge of the path P is crossed i D. I additio, as the graph G + P cotais K 6, as a subgraph with at least 6 1 crossigs, at most crossigs appear o the edges of G. Let us assume that o two edges of the subgraph G cross each other. The subdrawig of G iduced by D divides the plae ito oe quadragular ad two petagoal regios ad, as o edge of the path P is crossed, all vertices t i, i = 1,,..., are placed i oe of the petagoal regios. Assume ow a subdrawig D(F i ) of the subgraph F i = G T i. The edges of T i joiig the vertex t i with the vertices of G o the boudary of the cosidered petagoal regio divide this regio ito five ew regios. A vertex t j, j i, must be placed i oe of them. Oe ca easy verify that the edges of T j joiig t j with the vertices of G o the boudary of the petagoal regio of D(G) must cross T i at least four times. Hece, cr D (T i,t j ) 4 for 1 ISSN (prit) c 01 FEI TUKE Uautheticated Dowload Date 1/15/14 3:08 AM ISSN (olie)

4 Acta Electrotechica et Iformatica, Vol. 1, No. 3, i, j = 1,,...,, i j. Moreover, as G is ot outerplaar graph, every T i crosses G at least oce ad therefore, at least 4 ( ) + > crossigs appear i D. Fig. 3 The cosidered drawig of G T 1. This cotradictio implies that the edges of G cross each other ad that at least oe subgraph T i, say T 1, does ot cross the edges of G. So, all vertices of G must appear o the boudary of oe regio i the view of the subdrawig of G, say outside G, ad the subdrawig of G T 1 iduced by D divides the plae i such a way that o the boudary of every regio outside G there are two vertices of G, see Fig. 3, i which the possible crossigs amog the edges of G are iside the disc bouded by dotted cycle. As o edge of P is crossed, all vertices t i, i =,3,...,, are placed outside G i D. It is easy to verify i Fig. 3 that every T i crosses G T 1 at least four times, but the case cr D (G T 1,T i ) = 4 forces that T i crosses G at least twice. Thus, as cr D (G) 0 ad o the edges of G there are at most crossigs, at least oe subgraph T i crosses G T 1 more tha four times. This implies that cr D (G + P ) cr D (K 6, 1 ) + cr D (G + T 1 ) + cr D (K 6, 1,G T 1 ) 1 1 > 6 +. This cotradictio completes the proof ( ) THE CROSSING NUMBER OF G +C N The graph G + C cotais both G + P ad G + D as subgraphs. Let us deote by C the cycle t 1 t...t t 1 of the graph G + C. For x = a,b,c,d,e, f, let T x deote the subgraph of D 6 +C iduced o the edges icidet with the vertex x. Oe ca easily see that G +C = G K 6, C = G ( f ) = G T x C. x=a ( ) T i C = (9) i=1 The proof of the mai result of this sectio is based o the ext lemma ad its corollary which appeared i [14]. Lemma 4.1. ( [14]) Let D be a good drawig of the graph D m + C, m, 3, i which o edge of C is crossed ad C does ot separate the other vertices of the graph. The, for all i, j = 1,,...,m, two differet subgraphs T i ad T j cross each other at least 1 times i D. Corollary 4.1. ( [14]) Let D be a good drawig of the graph D m +C, m, 3, i which the edges of C cross each other ad oe of r subgraphs T i 1,T i,...t i r, r m, crosses the edges of C. The, for all j,k = 1,,...,r, two differet subgraphs T i j ad T i k cross each other at least 1 times i D. Theorem 4.1. cr(g +C ) = for 3. Proof. I Fig. 1, it is possible to add the cycle C = t 1 t...t t 1 i such way that the edge t t +1 crosses the edge bc ad edge t 1 t crosses the edges d f ad ae. So, cr(g +C ) Assume that there is a drawig D of the graph G+C with at most crossigs. Theorem 3.1 implies that, i D, o edge of C is crossed more tha oce, otherwise deletig a edge with two crossigs from C results i the drawig 1 of the graph G + P with fewer tha crossigs. Theorem.1 implies that there are at most two crossigs o the edges of C. Moreover, as G + C = G K 6, C ad cr(k 6, ) = 6 1, i D there are at most + crossigs o the edges of G C. Assume first that the cycle C crosses G i D. As C has at most two crossigs, it separates a) oe vertex of degree two, or b) two vertices of degree two of G from the other ad, as every edge of C is crossed at most oce, the graph G separates the vertices of C. Moreover, cr D (C ) = 0. Usig (9) ad the assumptio cr D (G,C ) = we have: cr D (G +C ) = cr D (G) + cr D (K 6, ) + cr D (C ) + + cr D (K 6,,C ) + cr D (K 6,,G) + cr D (G,C ) = cr D (G) + cr D (K 6, ) + cr D (K 6,,G) +. I the case a), cr D (K 6, ) ( 5) 1 sice, by Corollary 4.1, at least ( 5 ) pairs of subgraphs T x ad T y, x,y {a,b,c,d,e, f }, cross each other at least 1 times. Similarly, i the case b), cr D (K 6, ) ( 4) sice at least ( 4 ) pairs of T x ad T y cross each other at least 1 times ad moreover, the rest two subgraphs i the other regio i the view of C cross each other at least 1 times. Suppose ow that cr D (G) = 0. Sice G is ot outerplaar graph, G is crossed by all subgraphs T i, i = 1,,...,, givig cr D (K 6,,G). Hece, for the case a) cr D (G + C ) ( 5) ad for the case b) cr D (G +C ) ( 4) Both cases cotradict the assumptio of the drawig D. This forces that cr D (G) 0 ad, as cr D (G,C ) =, the restrictio of at most = crossigs o the edges of G C requires that cr D (G,T i ) = 0 for at least oe subgraph T i. ISSN (prit) c 01 FEI TUKE Uautheticated Dowload Date 1/15/14 3:08 AM ISSN (olie)

5 36 Miimum Crossigs i Joi of Graphs with Paths ad Cycles So, i the subdrawig D(G), all vertices of G are placed o the boudary of oe, say outside, regio. Moreover, as G separates the vertices of C, at least oe vertex of C lies i the regio of D(G) with at most four vertices of G o its boudary. This implies that there exists T i such that cr D (G,T i ). So, cr D (G,K 6, ). Thus, for the case a) cr D (G +C ) 1 + ( 5) ad for the case b) cr D (G +C ) 1 + ( 4) This cotradictio with our assumptio implies that C does ot cross the subgraph G i D. If cr D (C ) 0, the 4 ad, i the view of the subdrawig of C, all vertices of G are placed i the uique regio with all vertices of C o its boudary. I this case, at least five subgraphs T x, x {a,b,c,d,e, f }, do ot cross C ad, by Corollary 4.1, cr D (K 6, ) ( 5) 1. For 4, the umber of crossigs i D is at least ( 5) This cotradicts our assumptio agai. So, cr D (C ) = 0 ad the subdrawig of C iduced by D divides the plae ito two regios ad all vertices of G are placed i oe of them. If oe of the subgraphs T x, x {a,b,c,d,e, f }, crosses C, the, by Lemma 4.1, i the drawig D there are at least ( 6) 1 crossigs, a cotradictio with the assumptio. If oly oe subgraph T x, x {a,b,c,d,e, f }, crosses C oce or twice, the usig the fact that G has at least oe iteral crossig, the umber of crossigs i D is at least ( 5) This exceeds the cosidered umber of crossigs agai. If two subgraphs T x ad T y, x,y {a,b,c,d,e, f }, cross C, both T x ad T y cross every T q, q {a,b,c,d,e, f }, q x,y, at least 1 times ad therefore, there are at least ( 4 ) crossigs i D. This is more tha for 4. The last possibility is that = 3 ad two subgraphs T x ad T y cross the cycle C 3 i the drawig D. Recall that for cr D (K 6,3,C ) = ad = 3 the iequality cr D (G +C 3 ) = cr D (G) + cr D (K 6,3 ) + cr D (K 6,3,G) + cr D (G) + cr D (K 6,3,G) = cr D (G) + cr D (K 6,3,G) + 8 (10) holds. Thus, for the cosidered drawig D with at most eleve crossigs we have that cr D (G) + cr D (K 6,3,G) 3. Assume first that cr D (G) = 0. The subdrawig of G iduced by D divides the plae ito oe quadragular ad two petagoal regios ad, as cr D (G,C 3 ) = 0, all vertices t i, i = 1,,3, are placed i oe of the petagoal regios. The simple modificatio of Lemma 4.1 implies that the edges of T 1, T, ad T 3 joiig the vertices t 1, t, ad t 3 with the vertices o the boudary of this petagoal regio cross each other at least ( 3) = 1 times. This cotradict the assumptio of the drawig. Now, if cr D (G) 0, the, by (10), cr D (G,K 6, ). This forces that at least oe subgraph T i, say T 1, does ot cross the edges of G. Without loss of geerality, assume that the vertex t 1 is placed outside G i the view of the subdrawig of G. I this case, the subdrawig of G T 1 iduced by D divides the plae i such a way that o the boudary of every regio outside G there are two vertices of G, see Fig. 3, i which the possible crossigs amog the edges of G are iside the disc bouded by dotted cycle. All other vertices t i, i =,3, must be placed i the same regio outside G. It is easy to see i Fig. 3 that the case cr D (T i,g T 1 ) = 4 forces cr D (T i,g) ad cr D (T j,g T 1 ) for j i. Similarly, the case cr D (T i,g T 1 ) = 5 forces cr D (T i,g) 1 ad cr D (T j,g T 1 ) 5 for j i. Thus, cr D (T 1,T T 3 ) 10 i all possible cases, which, together with cr D (K 6,3,C 3 ) = cotradicts the assumptio of at most eleve crossigs i the drawig of G +C 3. This completes the proof. ACKNOWLEDGEMENT The research was supported by the Slovak VEGA grat No. 1/0309/11. This work was also supported by the Slovak Research ad Developmet Agecy uder the cotract No. APVV REFERENCES [1] BEINEKE, L. W. RINGEISEN, R. 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6 Acta Electrotechica et Iformatica, Vol. 1, No. 3, [14] KLEŠČ, M.: The joi of graphs of crossig umbers, Electroic Notes i Discrete Mathematics 8 (007), [15] KLEŠČ, M.: The crossig umbers of joi of the special graph o six vertices with path ad cycle, Discrete Math. 310 (010), [16] KLEŠČ, M. KOCÚROVÁ, A.: The crossig umber of products of 5-vertex graphs with cycles, Discrete Math. 307 (007), [17] KLEŠČ, M. SCHRÖTTER, Š.: The crossig umbers of joi products of paths with graphs of order four, Discuss. Math. Graph Theory 31 (011), [18] KLEŠČ, M. SCHRÖTTER, Š.: The crossig umbers of joi of paths ad cycles with two graphs of order five, Combiatorial Algorithms, Spriger, LNCS, 715 (01), [19] SHAHROKI, F. SÝKORA, O. SZÉKELY, L. A. VRŤO, I.: Itersectio of Curves ad Crossig Number of C m C o Surfaces, Discrete & Computatioal Geometry 19 (1998), [0] TURÁN, P.: A ote of welcome, J. Graph Theory 1 (1977), 7 9. [1] WOODALL, D. R.: Cyclic-order graphs ad Zarakiewicz s crossig-umber cojecture, J. Graph Theory 17 (1993), [] ZARANKIEWICZ, K.: O a problem of P. Turá cocerig graphs, Fud. Math 41 (1954), Received July 3, 01, accepted September 4, 01 BIOGRAPHIES Mariá Klešč is Associated Professor at the Departmet of Mathematics ad Theoretical Iformatics, Faculty of Electrical Egieerig ad Iformatics, Techical Uiversity of Košice. He received his PhD degree i discrete mathematics i 1995 at P. J. Šafárik Uiversity i Košice. His scietific research is focusig o topological graph theory ad crossig umbers problems. Matúš Valo graduated i iformatics from Techical Uiversity of Košice ad cotiues the studies as a postgraduate studet. His scietific research is focusig o crossig umbers problems of graphs ad their algorithmic computatio. ISSN (prit) c 01 FEI TUKE Uautheticated Dowload Date 1/15/14 3:08 AM ISSN (olie)