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The Isomorphism Classes of All Generalized Petersen Graphs The Isomorphism Classes of All Generalized Petersen Graphs Ted Dobson Department of Mathematics & Statistics Mississippi State University [email protected] http://www2.msstate.edu/∼dobson/ Seminar, University of Primorska, May 9, 2011 Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs The Petersen graph is GP(5; 2): (0; 0) • •(1; 0) (0; 4) (0; 1) • (1; 4) • • • (1; 1) (1; 3) • (1; 2)• • • (0; 3) (0; 2) Figure: The Petersen graph is GP(5; 2) For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n; k) to be the graph with vertex set Z2 × Zn and edge set f(0; i)(0; i + 1); (0; i)(1; i); (1; i)(1; i + k): i 2 Zng Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs (0; 0) • •(1; 0) (0; 4) (0; 1) • (1; 4) • • • (1; 1) (1; 3) • (1; 2)• • • (0; 3) (0; 2) Figure: The Petersen graph is GP(5; 2) For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n; k) to be the graph with vertex set Z2 × Zn and edge set f(0; i)(0; i + 1); (0; i)(1; i); (1; i)(1; i + k): i 2 Zng The Petersen graph is GP(5; 2): Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n; k) to be the graph with vertex set Z2 × Zn and edge set f(0; i)(0; i + 1); (0; i)(1; i); (1; i)(1; i + k): i 2 Zng The Petersen graph is GP(5; 2): (0; 0) • •(1; 0) (0; 4) (0; 1) • (1; 4) • • • (1; 1) (1; 3) • (1; 2)• • • (0; 3) (0; 2) Figure: The Petersen graph is GP(5; 2) Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether \similar" type graphs are Tait colorable. (0; 0) • (0; 9) (0; 1) • • (1; 9) •(1; 0) • •(1; 1) (0; 8)• (1; 8) •(0; 2) • • (1; 2) (1; 7) • • (0; 7)• (1; 3) •(0; 3) (1; 6)• • (1; 5)• (1; 4) • • (0; 6) (0; 4) • (0; 5) Figure: The generalized Petersen graph GP(10; 4). Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs The Petersen graph is not Tait colorable, so it is reasonable to ask whether \similar" type graphs are Tait colorable. (0; 0) • (0; 9) (0; 1) • • (1; 9) •(1; 0) • •(1; 1) (0; 8)• (1; 8) •(0; 2) • • (1; 2) (1; 7) • • (0; 7)• (1; 3) •(0; 3) (1; 6)• • (1; 5)• (1; 4) • • (0; 6) (0; 4) • (0; 5) Figure: The generalized Petersen graph GP(10; 4). Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs (0; 0) • (0; 9) (0; 1) • • (1; 9) •(1; 0) • •(1; 1) (0; 8)• (1; 8) •(0; 2) • • (1; 2) (1; 7) • • (0; 7)• (1; 3) •(0; 3) (1; 6)• • (1; 5)• (1; 4) • • (0; 6) (0; 4) • (0; 5) Figure: The generalized Petersen graph GP(10; 4). Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether \similar" type graphs are Tait colorable. Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether \similar" type graphs are Tait colorable. (0; 0) • (0; 9) (0; 1) • • (1; 9) •(1; 0) • •(1; 1) (0; 8)• (1; 8) •(0; 2) • • (1; 2) (1; 7) • • (0; 7)• (1; 3) •(0; 3) (1; 6)• • (1; 5)• (1; 4) • • (0; 6) (0; 4) • (0; 5) Figure: The generalized Petersen graph GP(10; 4). Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs p9 £D £ D p ` p7 £ 8 D 7 Q J`9 Q£ D QJ p6 `0 £ QD `3 `6 J `8 £ D Q `1 Jp4Q p0X D p5 XX £ "" X ` p3 `2XX 4 " X £ X "`5 p1 XXX"£ p2 p0 • ` ` 0• •1 •` p7 2 • •v1 p p 8• `9 • 5 • • `8 p2 • • p `6• 1 •`4 `3• • ` p6• 5 • • p9 p3 • `7 Figure: The Desargues configuration and its Levi graph GP(10; 3) Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs their automorphism groups are known, and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs and exactly which contain Hamilton cycles are known for example. Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, Ted Dobson Mississippi State University The Isomorphism Classes of All Generalized Petersen Graphs Many famous graphs are generalized Petersen graphs: I GP(5; 2) is the Petersen graph I GP(4; 1) is the skeleton of the cube I GP(10; 3) is the Desargues graph I GP(10; 2) is the skeleton of the dodecahedron I GP(8; 3) is the M¨obius-Kantor graph The generalized Petersen graphs have received a great deal of attention - their automorphism groups are known, and exactly which contain Hamilton cycles are known for example.
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