Planar Emulators Conjecture Is Nearly True for Cubic Graphs
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1 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar Emulators Conjecture Is Nearly True for Cubic Graphs Petr Hliněný Faculty of Informatics, Masaryk University Botanická 68a, Brno, Czech Rep. hlineny@fi.muni.cz * Joint work with Martin Derka, Brno / Waterloo.
2 Small math miracle for start... Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic...
3 Small math miracle for start... Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic...
4 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Small math miracle for start... turning nonplanar into planar
5 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Small math miracle for start... turning nonplanar into planar
6 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 1 Definitions Motivation: Exploring the two graphs locally, we cannot see any difference...
7 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 1 Definitions Motivation: Exploring the two graphs locally, we cannot see any difference... A graph H is a cover of a graph G if there exists a pair of onto mappings (a projection) ϕ : V(H) V(G), ψ : E(H) E(G) such that ψ maps the edges incident with each vertex v in H bijectively onto the edges incident with ϕ(v) in G. H v e 3 e 2 e 1 ϕ(v) ψ(e 3 ) ψ(e 2 ) ψ(e 1 ) G
8 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 1 Definitions Motivation: Exploring the two graphs locally, we cannot see any difference... A graph H is a cover of a graph G if there exists a pair of onto mappings (a projection) ϕ : V(H) V(G), ψ : E(H) E(G) such that ψ maps the edges incident with each vertex v in H bijectively onto the edges incident with ϕ(v) in G. H v e 3 e 2 e 1 ϕ(v) ψ(e 3 ) ψ(e 2 ) ψ(e 1 ) G Remark. The edge ψ(uv) has always ends ϕ(u), ϕ(v), and hence only ϕ : V(H) V(G), the vertex projection, is enough to be specified for simple graphs.
9 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar covers We speak about a planar cover if H is a finite planar graph. ϕ(v 1 ) = ϕ(v 2 ) = v v 2 v v 1 H G = K 5
10 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar covers We speak about a planar cover if H is a finite planar graph. ϕ(v 1 ) = ϕ(v 2 ) = v v 2 v v 1 H G = K 5 Graph embedded in the projective plane has a double planar cover, via the universal covering map from the sphere onto the proj. plane.
11 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar emulators ϕ : V(H) V(G), an emulator vs. a cover:... map the edges inc. with v in H surjectively onto the edges inc. with ϕ(v) in G.
12 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar emulators ϕ : V(H) V(G), an emulator vs. a cover:... map the edges inc. with v in H surjectively onto the edges inc. with ϕ(v) in G. c 2 b 2 a 3 a 2 a b H b 3 c 3 c 1 b 1 a 1 c G = K 3
13 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar emulators ϕ : V(H) V(G), an emulator vs. a cover:... map the edges inc. with v in H surjectively onto the edges inc. with ϕ(v) in G. c 2 b 2 a 3 a 2 a b H b 3 c 3 c 1 b 1 a 1 c G = K 3 Could a planar emulator be more than a planar cover?
14 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Planar emulators ϕ : V(H) V(G), an emulator vs. a cover:... map the edges inc. with v in H surjectively onto the edges inc. with ϕ(v) in G. c 2 b 2 a 3 a 2 a b H b 3 c 3 c 1 b 1 a 1 c G = K 3 Could a planar emulator be more than a planar cover? Not really, at least until
15 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 2 Fellows planar emulator conjecture Conjecture 1 (Negami, 1988) A connected graph has a finite planar cover it embeds in the projective plane.
16 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 2 Fellows planar emulator conjecture Conjecture 1 (Negami, 1988) A connected graph has a finite planar cover it embeds in the projective plane. Fact. A planar cover is also a planar emulator. Why a planar emulator should be more than a planar cover?
17 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 2 Fellows planar emulator conjecture Conjecture 1 (Negami, 1988) A connected graph has a finite planar cover it embeds in the projective plane. Fact. A planar cover is also a planar emulator. Why a planar emulator should be more than a planar cover? We only use more edges takes us farther away from planarity!
18 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 2 Fellows planar emulator conjecture Conjecture 1 (Negami, 1988) A connected graph has a finite planar cover it embeds in the projective plane. Fact. A planar cover is also a planar emulator. Why a planar emulator should be more than a planar cover? We only use more edges takes us farther away from planarity! Until the end of 2008, most people perhaps considered planar emulators just as a curious redefinition of planar covers...
19 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 2 Fellows planar emulator conjecture Conjecture 1 (Negami, 1988) A connected graph has a finite planar cover it embeds in the projective plane. Fact. A planar cover is also a planar emulator. Why a planar emulator should be more than a planar cover? We only use more edges takes us farther away from planarity! Until the end of 2008, most people perhaps considered planar emulators just as a curious redefinition of planar covers... Conjecture 2 (Fellows, 1989) A connected graph has a finite planar emulator it has a finite planar cover.
20 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Slow progress, and a sudden surprise Long-term development around Negami s conjecture led to... Theorem 3 (A+N+F+H, since 1998) If K 1,2,2,2 had no finite planar cover, then Negami s conj. would be proved.
21 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Slow progress, and a sudden surprise Long-term development around Negami s conjecture led to... Theorem 3 (A+N+F+H, since 1998) If K 1,2,2,2 had no finite planar cover, then Negami s conj. would be proved.... and then...
22 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Slow progress, and a sudden surprise Long-term development around Negami s conjecture led to... Theorem 3 (A+N+F+H, since 1998) If K 1,2,2,2 had no finite planar cover, then Negami s conj. would be proved.... and then... Suddenly, Fellows conjecture falls down... Fact. The graph K 4,5 4K 2 has no finite planar cover. Theorem 4 (Rieck and Yamashita 2008) The graphs K 1,2,2,2 and K 4,5 4K 2 do have finite planar emulators!!!
23 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Slow progress, and a sudden surprise Long-term development around Negami s conjecture led to... Theorem 3 (A+N+F+H, since 1998) If K 1,2,2,2 had no finite planar cover, then Negami s conj. would be proved.... and then... Suddenly, Fellows conjecture falls down... Fact. The graph K 4,5 4K 2 has no finite planar cover. Theorem 4 (Rieck and Yamashita 2008) The graphs K 1,2,2,2 and K 4,5 4K 2 do have finite planar emulators!!! Now we know that the class of graphs having finite planar emulators is different from the class of graphs having finite planar covers, and different from the class of projective planar graphs, too. So, let us study this class...
24 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... K 4,5 4K 2 (A picture by Yamashita.)
25 3 What graphs do have planar emulators? Recall... K 3,3 K 3,3 K 5 K 3,3 K 5 K 5 B 3 C 2 C 7 D 1 D 4 D 9 D 12 D 17 E 6 E 11 E 19 E 20 E 27 F 4 F 6 G 1 K 3,5 K 4,5 4K 2 K 4,4 e K 7 C 4 D 3 E 5 F 1 K 1,2,2,2 B 7 C 3 C 4 D 2 E 2 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic...
26 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Among the projective list NO emulators (proved) the case of two disjoint k-graphs, E 20 K 3,5
27 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Among the projective list NO emulators (proved) E 20 K 3,5 the case of two disjoint k-graphs, a sporadic proof for K 3,5 extends as well (nontrivial), but none of the other proofs from covers works for emulators.
28 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Among the projective list NO emulators (proved) E 20 K 3,5 YES emulators the case of two disjoint k-graphs, a sporadic proof for K 3,5 extends as well (nontrivial), but none of the other proofs from covers works for emulators. all projective-planar graphs, but those are the trivial ones,
29 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Among the projective list NO emulators (proved) E 20 K 3,5 YES emulators the case of two disjoint k-graphs, a sporadic proof for K 3,5 extends as well (nontrivial), but none of the other proofs from covers works for emulators. all projective-planar graphs, but those are the trivial ones, K 1,2,2,2 and K 4,5 4K 2 by [Rieck and Yamashita, 2008],
30 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Among the projective list NO emulators (proved) E 20 K 3,5 YES emulators the case of two disjoint k-graphs, a sporadic proof for K 3,5 extends as well (nontrivial), but none of the other proofs from covers works for emulators. all projective-planar graphs, but those are the trivial ones, K 1,2,2,2 and K 4,5 4K 2 by [Rieck and Yamashita, 2008], C 4 and E 2 by [PH and Chimani, 2009], and hence consequently the whole rich family of K 1,2,2,2,
31 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Among the projective list NO emulators (proved) E 20 K 3,5 YES emulators the case of two disjoint k-graphs, a sporadic proof for K 3,5 extends as well (nontrivial), but none of the other proofs from covers works for emulators. all projective-planar graphs, but those are the trivial ones, K 1,2,2,2 and K 4,5 4K 2 by [Rieck and Yamashita, 2008], C 4 and E 2 by [PH and Chimani, 2009], and hence consequently the whole rich family of K 1,2,2,2, and K 7 C 4 and its whole family by [Klusáček, 2011].
32 Graphically Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic...
33 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 4 The cubic case Although we do not much understand the whole class of nonprojective planar-emulable graphs is it essentially finite or infinite?
34 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 4 The cubic case Although we do not much understand the whole class of nonprojective planar-emulable graphs is it essentially finite or infinite? it appears significant that no such cubic graph has been found.
35 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 4 The cubic case Although we do not much understand the whole class of nonprojective planar-emulable graphs is it essentially finite or infinite? it appears significant that no such cubic graph has been found. We can thus use this easier ground to perhaps train our techniques before attacking the full problem... Theorem. If a cubic nonprojective graph H has a finite planar emulator, then H is a planar expansion of one of the following two graphs:
36 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Proof: G 1 G 2 The two graphs [Glover and Huneke, 1975] two out of all six cubic obstructions for projective embeddability; while the other four have two disjoint k-graphs.
37 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Proof: G 1 G 2 The two graphs [Glover and Huneke, 1975] two out of all six cubic obstructions for projective embeddability; while the other four have two disjoint k-graphs. Any nonproj. planar emul. graph H must contain a subdiv. of those.
38 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Proof: G 1 G 2 The two graphs [Glover and Huneke, 1975] two out of all six cubic obstructions for projective embeddability; while the other four have two disjoint k-graphs. Any nonproj. planar emul. graph H must contain a subdiv. of those. Cubic H H = G i + bridges where all the bridge legs subdivide the edges of G i.
39 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Proof: G 1 G 2 The two graphs [Glover and Huneke, 1975] two out of all six cubic obstructions for projective embeddability; while the other four have two disjoint k-graphs. Any nonproj. planar emul. graph H must contain a subdiv. of those. Cubic H H = G i + bridges where all the bridge legs subdivide the edges of G i. Even a single bridge having legs on non-incident edges of G i two disjoint k-graphs or a K 3,5 minor. [computer] OK
40 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... All bridges trivial (i.e., on incident legs) H = G i + bridges
41 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... All bridges trivial (i.e., on incident legs) H = G i + bridges Assign each trivial bridge to its G i -vertex or edge...
42 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... All bridges trivial (i.e., on incident legs) H = G i + bridges Assign each trivial bridge to its G i -vertex or edge... Must handle overlapping conflict of bridges:
43 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... All bridges trivial (i.e., on incident legs) H = G i + bridges Assign each trivial bridge to its G i -vertex or edge... Must handle overlapping conflict of bridges: Any conflict another subdivision gives a non-trivial bridge.
44 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... All bridges trivial (i.e., on incident legs) H = G i + bridges Assign each trivial bridge to its G i -vertex or edge... Must handle overlapping conflict of bridges: Any conflict another subdivision gives a non-trivial bridge. Hence the assign. of bridges to vert./edges of G i is rigorous. OK
45 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Trivial bridges are local H = G i + bridges
46 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Trivial bridges are local H = G i + bridges Non-conflicting trivial bridges at any given vertex;
47 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Trivial bridges are local H = G i + bridges Non-conflicting trivial bridges at any given vertex; either planar expansion (left), or K 2,3 in the fragment (twice right),
48 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... Trivial bridges are local H = G i + bridges Non-conflicting trivial bridges at any given vertex; either planar expansion (left), or K 2,3 in the fragment (twice right), End of proof. two disjoint k-graphs. OK
49 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!).
50 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!). Any idea for a new hypothesis concerning (non-cubic) graphs?
51 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!). Any idea for a new hypothesis concerning (non-cubic) graphs? Prove that the required fold number is finite for planar emulators?
52 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!). Any idea for a new hypothesis concerning (non-cubic) graphs? Prove that the required fold number is finite for planar emulators? And, of course, do not forget still open Negami s conjecture!
53 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!). Any idea for a new hypothesis concerning (non-cubic) graphs? Prove that the required fold number is finite for planar emulators? And, of course, do not forget still open Negami s conjecture! Lastly, a structural question for which graph class G; { } G-emulable/coverable = G?
54 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!). Any idea for a new hypothesis concerning (non-cubic) graphs? Prove that the required fold number is finite for planar emulators? And, of course, do not forget still open Negami s conjecture! Lastly, a structural question for which graph class G; { } G-emulable/coverable = G? Holds true for G = outerplanar (so, k-outerplanar?),
55 Petr Hliněný, Eurocomb, Pisa, / 16 Planar Emulators Conjecture for Cubic... 5 Conclusion Resolve the two cubic graphs (everything else is now finished!). Any idea for a new hypothesis concerning (non-cubic) graphs? Prove that the required fold number is finite for planar emulators? And, of course, do not forget still open Negami s conjecture! Lastly, a structural question for which graph class G; { } G-emulable/coverable = G? Holds true for G = outerplanar (so, k-outerplanar?), Negami true for G =projective (and so false with emul.), other classes, e.g., G = other nonorientable surface?
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