- Email: [email protected]
Infinite families of crossing-critical graphs with a given crossing number
Discrete Mathematics 48 (1984) 129-132 North-Holland
129
NOTE
INFINITE FAMILIES OF CROSSING-CRITICAL WITH A GIVEN CROSSING NUMBER
GRAPHS
Jozef ~ I t L / ~
Departmentof Telecommunications, EF SVS"~T,Vazovova 5, 812 19 Bratislava, Czechoslovakia Received 16 July 1982 A graph is crossing-critical if deleting any edge decreases its crossing number on the plane. It is proved that, for any n >~3, there is an infinite family of 3-connected crossing-critical graphs with crossing number n.
1. Introduction Let s be a topological invariant of graphs, i.e. an integer valued function for which s(G)= s(H) whenever the graphs G, H are homeomorphic. W e shall say that s is a m o n o t o n o u s invariant of graphs if s(G-e)<<.s(G) for every graph G and every edge e of G. Further, a graph G is said to be s-critical if s ( G - e ) < s(G) for every edge e of G. Suppose that s is a m o n o t o n o u s topological inx,ariant of graphs. It is easy to see that for each integer n there is a family of 'forbidden subgraphs' F,(s) such that s(G)<~n if and only if G does not contain a subgraph h o m e o m o r p h i c to a m e m b e r of F, (s). In order to minimize the cardinality of F, (s) we shall require in addition that each graph of F,(s) is s-critical and no two graphs of F~(s) are homeomorphic. It would be unreasonable to expect to find a satisfactory characterization of the families F,(s) for every n, but there is an interesting and often discussed particular question: For which s and n is F,(s) a finite set? T h e genus and the (planar) crossing n u m b e r are examples of m o n o t o n o u s topological invariants of graphs. As regards the first one, i.e. if s(G)= 3,(G), the (orientable or nonorientable) genus of G, there is a well known conjecture of Erd6s that F,(3,) is a finite set for each n/> 1 (recall that for n = 0 this follows f r o m the classical Kuratowski's t h e o r e m [5]). U p to the present time this conjecture has been verified only for the nonorientable case n = 1 (cf. [2]). T h e purpose of this p a p e r is to show that in the second case, if s(G)=cr(G), the planar crossing n u m b e r of G, the set F, (cr) is infinite for each n I> 2, in contrast with the above conjecture. In fact, we p r o v e a stronger result concerning the n u m b e r of 0012-365x/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
130
J. ~ir6h
crossing-critical (s-critical for s = cr) graphs with a given crossing number (Section 2). It may occur that a m e m b e r G of F,(cr) contains several parallel paths Pi connecting two its vertices u, v such that each vertex of Pi different from u, v has degree 2 in G. There is therefore no reason to restrict our investigation only to simple graphs, i.e. we allow two vertices of our graph to be joined by several parallel edges. For definitions of an immersion and a minimal immersion of a graph on the plane see Kainen and White [4]. The concept of a minimal immersion corresponds to that of an optimal drawing of a graph on the plane (cf. e.g. [l]). Other terminology is essentially the same as that of Harary [3].
2. The construction
Let G be a graph. W e shall say that Go is the edge-labeled graph associated with G if both G, Go have the same sets of vertices and the following condition holds: Two vertices a, b are joined in Go by a single edge labeled n if and only if vertices a, b are joined in G by exactly n parallel edges. Let Go be the edge-labeled graph associated with a graph G. D e n o t e by P the plane and consider an immersion f: Go --~ P. By the multiplicity of a crossing C in the 'drawing' f(Go) on the plane we simply mean the product of labels of those two edges of Go which are 'responsible' for the crossing C in f(Go). Define the multiple crossing number crm(Go) to be the minimum sum of multiplicities of crossings over all immersions Go --~ P. Clearly crm(Go) = cr(G). Theorem. For any n >t 3 there is an infinite f a m i l y o f 3-connected crossing-critical graphs with crossing n u m b e r n. Proof. For n>13 and k ~ l let G = G ( n , k ) be the graph depicted in Fig. 1. Obviously G is 3-connected and cr(G)~< n for any n, k. We want to establish the reverse inequality. Assume the contrary, i.e. that cr(G)<~ n - 1. Let Go be the edge-labeled graph associated with G. Denote by M the set of all edges of Go incident to a or b (Fig. 1). Consider two subgraphs H, K of Go, H induced by M and paths vw, t t t ! I t ! 1)Vl'02''" l)kS, 1)UUlUa'''Ukt, K induced by M and paths v'w, v ~91~)2.-.l)kt , v ' u u l u 2 • - • uks. Note that both H, K are subdivisions of the graph K3.3. Further, every edge of H and K is labeled 1, n - I or n, and no edge labeled n - 1 is common to H and K. Now let f : G o - - > P be an optimal drawing of Go, i.e. an immersion for which the sum of multiplicities of crossings is minimal. If f ( H ) has a crossing of multiplicity n - l , then f ( K ) must have a different crossing, contradicting crm(Go) ~ n - l. Therefore all crossings in f ( H ) and f ( K ) must have multiplicity 1. Since the only edges labeled 1 that are common to H and K are all incident to the same vertex, f ( H ) and f ( K ) have no common crossings.
Infinite families of crossing-critical graphs
131
uk °o
u2 aQ o
V,
uk
..........................
n - 2 parallel edges
Fig. 1. The graph G = G(n, k~. Let L be another subdivision of K3.3 in Go induced by M and paths u v u l v l u 2 v 2 " " " UkVkS, UV'U~V'lU~V~" " " U~V'kt and uw. T h e drawing f ( L ) certainly contains at least one crossing of multiplicity at least n - 2. If n >/4, then the last crossing is different f r o m those of f ( H ) and [(K), contradicting our assumption that crm(G0) ~< n - 1. It remains to investigate the case n = 3. Observe that in this case again H and L have no c o m m o n edges labeled 1 except those incident to b; a similar result holds for K and L. Consequently, f ( L ) contains a crossing different f r o m those of f ( H ) and f ( K ) , whence c r m ( G 0 ) ~ 3, a contradiction. Having established the equality cr(G) --- n, we only need to prove that all graphs G = G ( n , k) are indeed crossing-critical. But this can be easily checked looking at Fig. I O u r t h e o r e m follows. Corollary. For a n y natural numbers n >I 3 and m there is a 3-connected crossingcritical graph G with cr(G) = n w h i c h contains at least m different subdivisions o f K3.3.
132 Proof. T h e graph G =
J. ,{ird~t
G(n, m)
constructed a b o v e has the desired property.
For example, 3-connected crossing-critical graphs with crossing n u m b e r 3 may contain an arbitrarily large n u m b e r of subdivisions of K3.3 (having crossing n u m b e r 1)!
3. Remarks It follows f r o m the above t h e o r e m that for every n ~>2 the set F,(cr) is infinite. However, it is not true that Fn(cr) consists only of crossing-critical graphs G for which c r ( G ) = n + 1. T h e product of two 3-cycles C3 × (73 is a nice example of a graph which belongs to Fl(cr) but has crossing n u m b e r 3. As can be seen f r o m Fig. 1, each of the crossing-critical graphs G(n, k) has the property that each edge of G(n, k) belongs to a subdivision of /(3.3 in G(n, k). Nevertheless, there is an infinite n u m b e r of crossing-critical graphs G containing edges which belong to no subdivisions of/(3.3 o r / ( 5 , the Kuratowski subgraphs, of a (cf. [6]). It is a consequence of Kuratowski's t h e o r e m [5] that there are only. two 3-connected crossing-critical graphs with crossing n u m b e r 1. Thus, as regards the n u m b e r of 3-connected crossing-critical graphs with a given crossing n u m b e r n, the only one case that remains unsolved is the case n = 2. So far we have listed some twenty 3-connected crossing-critical graphs with crossing n u m b e r 2 and believe that the n u m b e r of all such graphs is finite.
Conjecture. T h e r e are only finitely m a n y 3-connected crossing-critical graphs with crossing n u m b e r 2.
References [1] P. Erd6s and R.K. Guy, Crossing number problems, Amer. Math. Monthly 80 (1973) 52-58. [2] H.H. Glover and J.P. Huneke, There are finitely many Kuratowski graphs for the projective plane, in: J.A. Bondy and U.S.R. Murty, eds., Graph Theory and Related Topics (Academic press, New York, 1979) 201-206. [3] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [4] P.C. Kainen and A.T. White, On stable crossing numbers, J. Graph Theory 2 (1978) 181-187. [5] K. Kuratowski, Sur le probl6me des courbes gauches en topologie, Fund. Math. 15 (1930) 271-283. [6] J. ~irfi~, v-function, v-critical edges and v-critical graphs, in: Proc. Colloq. Eger, 1981 (to appear).
129
NOTE
INFINITE FAMILIES OF CROSSING-CRITICAL WITH A GIVEN CROSSING NUMBER
GRAPHS
Jozef ~ I t L / ~
Departmentof Telecommunications, EF SVS"~T,Vazovova 5, 812 19 Bratislava, Czechoslovakia Received 16 July 1982 A graph is crossing-critical if deleting any edge decreases its crossing number on the plane. It is proved that, for any n >~3, there is an infinite family of 3-connected crossing-critical graphs with crossing number n.
1. Introduction Let s be a topological invariant of graphs, i.e. an integer valued function for which s(G)= s(H) whenever the graphs G, H are homeomorphic. W e shall say that s is a m o n o t o n o u s invariant of graphs if s(G-e)<<.s(G) for every graph G and every edge e of G. Further, a graph G is said to be s-critical if s ( G - e ) < s(G) for every edge e of G. Suppose that s is a m o n o t o n o u s topological inx,ariant of graphs. It is easy to see that for each integer n there is a family of 'forbidden subgraphs' F,(s) such that s(G)<~n if and only if G does not contain a subgraph h o m e o m o r p h i c to a m e m b e r of F, (s). In order to minimize the cardinality of F, (s) we shall require in addition that each graph of F,(s) is s-critical and no two graphs of F~(s) are homeomorphic. It would be unreasonable to expect to find a satisfactory characterization of the families F,(s) for every n, but there is an interesting and often discussed particular question: For which s and n is F,(s) a finite set? T h e genus and the (planar) crossing n u m b e r are examples of m o n o t o n o u s topological invariants of graphs. As regards the first one, i.e. if s(G)= 3,(G), the (orientable or nonorientable) genus of G, there is a well known conjecture of Erd6s that F,(3,) is a finite set for each n/> 1 (recall that for n = 0 this follows f r o m the classical Kuratowski's t h e o r e m [5]). U p to the present time this conjecture has been verified only for the nonorientable case n = 1 (cf. [2]). T h e purpose of this p a p e r is to show that in the second case, if s(G)=cr(G), the planar crossing n u m b e r of G, the set F, (cr) is infinite for each n I> 2, in contrast with the above conjecture. In fact, we p r o v e a stronger result concerning the n u m b e r of 0012-365x/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
130
J. ~ir6h
crossing-critical (s-critical for s = cr) graphs with a given crossing number (Section 2). It may occur that a m e m b e r G of F,(cr) contains several parallel paths Pi connecting two its vertices u, v such that each vertex of Pi different from u, v has degree 2 in G. There is therefore no reason to restrict our investigation only to simple graphs, i.e. we allow two vertices of our graph to be joined by several parallel edges. For definitions of an immersion and a minimal immersion of a graph on the plane see Kainen and White [4]. The concept of a minimal immersion corresponds to that of an optimal drawing of a graph on the plane (cf. e.g. [l]). Other terminology is essentially the same as that of Harary [3].
2. The construction
Let G be a graph. W e shall say that Go is the edge-labeled graph associated with G if both G, Go have the same sets of vertices and the following condition holds: Two vertices a, b are joined in Go by a single edge labeled n if and only if vertices a, b are joined in G by exactly n parallel edges. Let Go be the edge-labeled graph associated with a graph G. D e n o t e by P the plane and consider an immersion f: Go --~ P. By the multiplicity of a crossing C in the 'drawing' f(Go) on the plane we simply mean the product of labels of those two edges of Go which are 'responsible' for the crossing C in f(Go). Define the multiple crossing number crm(Go) to be the minimum sum of multiplicities of crossings over all immersions Go --~ P. Clearly crm(Go) = cr(G). Theorem. For any n >t 3 there is an infinite f a m i l y o f 3-connected crossing-critical graphs with crossing n u m b e r n. Proof. For n>13 and k ~ l let G = G ( n , k ) be the graph depicted in Fig. 1. Obviously G is 3-connected and cr(G)~< n for any n, k. We want to establish the reverse inequality. Assume the contrary, i.e. that cr(G)<~ n - 1. Let Go be the edge-labeled graph associated with G. Denote by M the set of all edges of Go incident to a or b (Fig. 1). Consider two subgraphs H, K of Go, H induced by M and paths vw, t t t ! I t ! 1)Vl'02''" l)kS, 1)UUlUa'''Ukt, K induced by M and paths v'w, v ~91~)2.-.l)kt , v ' u u l u 2 • - • uks. Note that both H, K are subdivisions of the graph K3.3. Further, every edge of H and K is labeled 1, n - I or n, and no edge labeled n - 1 is common to H and K. Now let f : G o - - > P be an optimal drawing of Go, i.e. an immersion for which the sum of multiplicities of crossings is minimal. If f ( H ) has a crossing of multiplicity n - l , then f ( K ) must have a different crossing, contradicting crm(Go) ~ n - l. Therefore all crossings in f ( H ) and f ( K ) must have multiplicity 1. Since the only edges labeled 1 that are common to H and K are all incident to the same vertex, f ( H ) and f ( K ) have no common crossings.
Infinite families of crossing-critical graphs
131
uk °o
u2 aQ o
V,
uk
..........................
n - 2 parallel edges
Fig. 1. The graph G = G(n, k~. Let L be another subdivision of K3.3 in Go induced by M and paths u v u l v l u 2 v 2 " " " UkVkS, UV'U~V'lU~V~" " " U~V'kt and uw. T h e drawing f ( L ) certainly contains at least one crossing of multiplicity at least n - 2. If n >/4, then the last crossing is different f r o m those of f ( H ) and [(K), contradicting our assumption that crm(G0) ~< n - 1. It remains to investigate the case n = 3. Observe that in this case again H and L have no c o m m o n edges labeled 1 except those incident to b; a similar result holds for K and L. Consequently, f ( L ) contains a crossing different f r o m those of f ( H ) and f ( K ) , whence c r m ( G 0 ) ~ 3, a contradiction. Having established the equality cr(G) --- n, we only need to prove that all graphs G = G ( n , k) are indeed crossing-critical. But this can be easily checked looking at Fig. I O u r t h e o r e m follows. Corollary. For a n y natural numbers n >I 3 and m there is a 3-connected crossingcritical graph G with cr(G) = n w h i c h contains at least m different subdivisions o f K3.3.
132 Proof. T h e graph G =
J. ,{ird~t
G(n, m)
constructed a b o v e has the desired property.
For example, 3-connected crossing-critical graphs with crossing n u m b e r 3 may contain an arbitrarily large n u m b e r of subdivisions of K3.3 (having crossing n u m b e r 1)!
3. Remarks It follows f r o m the above t h e o r e m that for every n ~>2 the set F,(cr) is infinite. However, it is not true that Fn(cr) consists only of crossing-critical graphs G for which c r ( G ) = n + 1. T h e product of two 3-cycles C3 × (73 is a nice example of a graph which belongs to Fl(cr) but has crossing n u m b e r 3. As can be seen f r o m Fig. 1, each of the crossing-critical graphs G(n, k) has the property that each edge of G(n, k) belongs to a subdivision of /(3.3 in G(n, k). Nevertheless, there is an infinite n u m b e r of crossing-critical graphs G containing edges which belong to no subdivisions of/(3.3 o r / ( 5 , the Kuratowski subgraphs, of a (cf. [6]). It is a consequence of Kuratowski's t h e o r e m [5] that there are only. two 3-connected crossing-critical graphs with crossing n u m b e r 1. Thus, as regards the n u m b e r of 3-connected crossing-critical graphs with a given crossing n u m b e r n, the only one case that remains unsolved is the case n = 2. So far we have listed some twenty 3-connected crossing-critical graphs with crossing n u m b e r 2 and believe that the n u m b e r of all such graphs is finite.
Conjecture. T h e r e are only finitely m a n y 3-connected crossing-critical graphs with crossing n u m b e r 2.
References [1] P. Erd6s and R.K. Guy, Crossing number problems, Amer. Math. Monthly 80 (1973) 52-58. [2] H.H. Glover and J.P. Huneke, There are finitely many Kuratowski graphs for the projective plane, in: J.A. Bondy and U.S.R. Murty, eds., Graph Theory and Related Topics (Academic press, New York, 1979) 201-206. [3] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [4] P.C. Kainen and A.T. White, On stable crossing numbers, J. Graph Theory 2 (1978) 181-187. [5] K. Kuratowski, Sur le probl6me des courbes gauches en topologie, Fund. Math. 15 (1930) 271-283. [6] J. ~irfi~, v-function, v-critical edges and v-critical graphs, in: Proc. Colloq. Eger, 1981 (to appear).