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K6 and icosahedron minors in 5-connected projective planar graphs
K6
and icosahedron minors in 5-connected projective planar graphs ? Gasper Fijavz 1
Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Abstract We show that every 5-connected graph admitting an embedding into the projective plane with face-width at least 3 contains K6 as a minor. Further we nd a simple proof that every 5-connected planar graph contracts to icosahedron. We adapt the proof and nd a simple face-width type condition such that every 5-connected graph embedded in an arbitrary surface satisfying the condition contracts to icosahedron. Key words: projective-planar graphs, 5-connected graphs, graph minors, face-width.
1 Introduction and motivation The work presented in this text was motivated by a paper of Halin and Jung [2] and a paper of Mader [3] on minimal structures in k-connected graphs. It is obvious that every 3-connected graph contains K4 as a minor. A well known theorem from Wagner [6] states that if a 4-connected graph G does not contain K5 as a minor then G is planar. It is easy to see that every 4connected planar graph G contracts to K2 2 2, the octahedron. How? Suppose we have an embedding of G. Pick an arbitrary vertex v which will also serve as a vertex of the octahedron minor. Let C be the facial cycle of the embedding of G v. Now G C v is connected since C is a facial cycle in a 3-connected graph G v and therefore does not separate the remaining graph. We contract G C v to a vertex which is antipodal to v in K2 2 2 minor and obtain the ; ;
; ;
This work was supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1-0502-0101-98. 1 E-mail: gasper. [email protected] ?
Preprint submitted to Elsevier Preprint
15 May 2000
remaining four vertices from C . Hence, every 4-connected graph contains K5 or K2 2 2 as a minor. Maharry [4] gives an alternative approach excluding the octahedron. ; ;
The analogy for 5-connected graphs is not as straightforward as one could hope for. We only studied the extremal two cases, namely, K6 as the complete 5-connected graph, and the icosahedron as the planar one.
2 K6 in projective planar graphs The following theorem has been obtained recently.
Theorem 1 (Fijavz and Mohar [1]) Let G be a 5-connected graph embed-
ded into the projective plane with face-width at least 3. Then K6 is a minor of G.
First we argue that the assumptions are somehow best possible. There is only one embedding of K6 in the projective plane (up to automorphism) and it is of face-width 3. Since face-width is minor-monotone, K6 cannot be a minor of a graph embedded in the projective plane with face-width 2. Further we were able to nd a family of 4-connected projective planar graphs with minimum degree 5 (and face-width 3) which do not contain K6 as a minor. They are constructed using a projective embedding of K3 3 K1. ;
Suppose we have a wheel with 5 rim vertices and a cycle of length 5. If we identify rim vertices of a wheel with those of a cycle in a suitable manner we obtain a K6 . This is the strategy used for constructing a K6 minor in a graph satisfying the assumptions of the theorem. We choose a suitable vertex v of degree 5. Its neighbors form a 5-separating-face-chain (a sequence of 5 consecutively incident vertices and faces which is `nullhomotopic' and contains v in its interior). We choose a maximum 5-separating-face-chain containing v in its interior and proceed to nd a wheel minor in its interior and a matching 5-cycle minor in the remaining part of the graph. We refrain from giving the technical details.
3 Icosahedron as a minor It was shown in [3] that every planar graph with minimum degree 5 contains icosahedron as a minor. Here we give a very simple argument for a weaker result, that every 5-connected planar graph contains icosahedron as a minor. On the other hand, this argument can be generalized to other surfaces. 2
Suppose that G is an internally 6-connected planar graph (i.e. it is 5-connected and every 5 separator is exactly the neighborhood of a 5-valent vertex). We choose an arbitrary vertex v, we call it the initial vertex, and look at its second neighborhood N , i.e. the vertices (and edges and faces) whose `facial distance' from v is at most 2. It is easy to see that this second neighborhood contracts to icosahedron minus a vertex. The remaining vertex, which is to be antipodal to v in the icosahedron minor, is then obtained by contracting G N . If G is not internally 6-connected we choose a `big' 5 separator S (it should separate more than just a single vertex from the rest of the graph) with minimum number of vertices in its interior (we assume that G is embedded in 2-sphere and choose interior arbitrarily). We show that there are at least 6 vertices lying inside and that there is a vetex w at `facial distance' at least 2 from S. We then choose w as an initial vertex for icosahedron construction. Let G be a 5-connected graph embedded into a surface of higher genus and let v be a vertex of G. Let fw(v) denote the minimal size of jC \ Gj over all noncontractible curves C which pass through v. The face-width of G, fw(G) = minv2V G fw(v). On the other hand we de ne FW(G) = maxv2V G fw(v). It is easy to see that FW is minor-monotone. Suppose that fw(v) 6. Then we use v as an initial vertex for constructing the icosahedron. We are successful unless v is a vertex of some `nullhomotopic' 5-separator that contains a lot of vertices in its interior. Among all such 5-separators let S be the one with minimum number of vertices in its interior and let u be a vertex in the interior of S which is at the maximum facial distance from the vertices of S . It is easy to see that fw(u) fw(v) and that u can be chosen for the initial vertex. Hence: (
)
(
)
Theorem 2 The icosahedron is contained as a minor in every 5-connected G with FW(G) 6. Using the fact that every noncontractible curve in the projective plane is nonseparating we have an even stronger result.
Theorem 3 Let G be a 5-connected projective planar graph and FW(G) 5. Then icosahedron is a minor in G. References [1] G. Fijavz, B. Mohar { (2000).
K -minors in projective planar graphs, submitted 6
[2] R. Halin, H. A. Jung { U ber Minimalstrukturen von Graphen, insbesondere von n-fach zusammenhangenden Graphen, Math. Ann. 152 (1963), 75{94.
3
[3] W. Mader { Homomorphiesatze fur Graphen, Math. Ann. 178 (1968), 154{ 168. [4] J. Maharry { An excluded minor theorem for the octahedron, J. Graph Theory 31, (1999), 95{100, [5] N. Robertson, P. D. Seymour { Graph minors. VI. Disjoint paths across a disc, J. Combin. Theory Ser. B 41 (1986), 115{138. [6] K. Wagner { U ber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), 570{590.
4
and icosahedron minors in 5-connected projective planar graphs ? Gasper Fijavz 1
Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Abstract We show that every 5-connected graph admitting an embedding into the projective plane with face-width at least 3 contains K6 as a minor. Further we nd a simple proof that every 5-connected planar graph contracts to icosahedron. We adapt the proof and nd a simple face-width type condition such that every 5-connected graph embedded in an arbitrary surface satisfying the condition contracts to icosahedron. Key words: projective-planar graphs, 5-connected graphs, graph minors, face-width.
1 Introduction and motivation The work presented in this text was motivated by a paper of Halin and Jung [2] and a paper of Mader [3] on minimal structures in k-connected graphs. It is obvious that every 3-connected graph contains K4 as a minor. A well known theorem from Wagner [6] states that if a 4-connected graph G does not contain K5 as a minor then G is planar. It is easy to see that every 4connected planar graph G contracts to K2 2 2, the octahedron. How? Suppose we have an embedding of G. Pick an arbitrary vertex v which will also serve as a vertex of the octahedron minor. Let C be the facial cycle of the embedding of G v. Now G C v is connected since C is a facial cycle in a 3-connected graph G v and therefore does not separate the remaining graph. We contract G C v to a vertex which is antipodal to v in K2 2 2 minor and obtain the ; ;
; ;
This work was supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1-0502-0101-98. 1 E-mail: gasper. [email protected] ?
Preprint submitted to Elsevier Preprint
15 May 2000
remaining four vertices from C . Hence, every 4-connected graph contains K5 or K2 2 2 as a minor. Maharry [4] gives an alternative approach excluding the octahedron. ; ;
The analogy for 5-connected graphs is not as straightforward as one could hope for. We only studied the extremal two cases, namely, K6 as the complete 5-connected graph, and the icosahedron as the planar one.
2 K6 in projective planar graphs The following theorem has been obtained recently.
Theorem 1 (Fijavz and Mohar [1]) Let G be a 5-connected graph embed-
ded into the projective plane with face-width at least 3. Then K6 is a minor of G.
First we argue that the assumptions are somehow best possible. There is only one embedding of K6 in the projective plane (up to automorphism) and it is of face-width 3. Since face-width is minor-monotone, K6 cannot be a minor of a graph embedded in the projective plane with face-width 2. Further we were able to nd a family of 4-connected projective planar graphs with minimum degree 5 (and face-width 3) which do not contain K6 as a minor. They are constructed using a projective embedding of K3 3 K1. ;
Suppose we have a wheel with 5 rim vertices and a cycle of length 5. If we identify rim vertices of a wheel with those of a cycle in a suitable manner we obtain a K6 . This is the strategy used for constructing a K6 minor in a graph satisfying the assumptions of the theorem. We choose a suitable vertex v of degree 5. Its neighbors form a 5-separating-face-chain (a sequence of 5 consecutively incident vertices and faces which is `nullhomotopic' and contains v in its interior). We choose a maximum 5-separating-face-chain containing v in its interior and proceed to nd a wheel minor in its interior and a matching 5-cycle minor in the remaining part of the graph. We refrain from giving the technical details.
3 Icosahedron as a minor It was shown in [3] that every planar graph with minimum degree 5 contains icosahedron as a minor. Here we give a very simple argument for a weaker result, that every 5-connected planar graph contains icosahedron as a minor. On the other hand, this argument can be generalized to other surfaces. 2
Suppose that G is an internally 6-connected planar graph (i.e. it is 5-connected and every 5 separator is exactly the neighborhood of a 5-valent vertex). We choose an arbitrary vertex v, we call it the initial vertex, and look at its second neighborhood N , i.e. the vertices (and edges and faces) whose `facial distance' from v is at most 2. It is easy to see that this second neighborhood contracts to icosahedron minus a vertex. The remaining vertex, which is to be antipodal to v in the icosahedron minor, is then obtained by contracting G N . If G is not internally 6-connected we choose a `big' 5 separator S (it should separate more than just a single vertex from the rest of the graph) with minimum number of vertices in its interior (we assume that G is embedded in 2-sphere and choose interior arbitrarily). We show that there are at least 6 vertices lying inside and that there is a vetex w at `facial distance' at least 2 from S. We then choose w as an initial vertex for icosahedron construction. Let G be a 5-connected graph embedded into a surface of higher genus and let v be a vertex of G. Let fw(v) denote the minimal size of jC \ Gj over all noncontractible curves C which pass through v. The face-width of G, fw(G) = minv2V G fw(v). On the other hand we de ne FW(G) = maxv2V G fw(v). It is easy to see that FW is minor-monotone. Suppose that fw(v) 6. Then we use v as an initial vertex for constructing the icosahedron. We are successful unless v is a vertex of some `nullhomotopic' 5-separator that contains a lot of vertices in its interior. Among all such 5-separators let S be the one with minimum number of vertices in its interior and let u be a vertex in the interior of S which is at the maximum facial distance from the vertices of S . It is easy to see that fw(u) fw(v) and that u can be chosen for the initial vertex. Hence: (
)
(
)
Theorem 2 The icosahedron is contained as a minor in every 5-connected G with FW(G) 6. Using the fact that every noncontractible curve in the projective plane is nonseparating we have an even stronger result.
Theorem 3 Let G be a 5-connected projective planar graph and FW(G) 5. Then icosahedron is a minor in G. References [1] G. Fijavz, B. Mohar { (2000).
K -minors in projective planar graphs, submitted 6
[2] R. Halin, H. A. Jung { U ber Minimalstrukturen von Graphen, insbesondere von n-fach zusammenhangenden Graphen, Math. Ann. 152 (1963), 75{94.
3
[3] W. Mader { Homomorphiesatze fur Graphen, Math. Ann. 178 (1968), 154{ 168. [4] J. Maharry { An excluded minor theorem for the octahedron, J. Graph Theory 31, (1999), 95{100, [5] N. Robertson, P. D. Seymour { Graph minors. VI. Disjoint paths across a disc, J. Combin. Theory Ser. B 41 (1986), 115{138. [6] K. Wagner { U ber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), 570{590.
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