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Edge Maps and Isomorphisms of Graphs
Annals of Discrete Mathematics 20 (1984) 327-328 North-Holland
EDGE MAPS AND ISOMORPHISMS OF GRAPHS
A.K. KELMANS Profsouznaya Str. 130, 117321 Moscow, U.S.S.R.
Let G be an undirected graph and EG denote the edge-set of G. The union of n 2 1 chains of G is called an n-skein [l] if they have the common end-points (the terminals of the skein) and no pair of them has an inner point in common. Let S, ( G )denote the set of n-skeins of G. A 1-1 map e :EG .--, EF is called an n-skein isomorphism of G onto F if A ES,(G) @ e(A)€S.(F). In [l] the following result is obtained. Theorem 1. Suppose that G and F are (n + 1)-vertex connected, n 2 2. Then G and F a r e isomorphic if and only if they are n-skein isomorphic, and any n-skein isomorphism of G onto F is induced by a n isomorphism of G onto E
The same assertion for n-skeins in n-connected graphs is wrong, as Whitney [2] showed for n = 2. Also, it is easy to see [l] that every permutation of the edge-set of K4 is a 3-skein automorphism of K4, but clearly not every such map is induced by an automorphism of K,. In [l] we read: ". . . we were not able to find more complicated examples in the case n = 3, nor could we find for n 3 4 an example of an n-skein isomorphism of an n-vertex connected graph which was not induced by an isomorphism". Here we see that no other example of this kind exists for 3-connected graphs. Theorem 2. Let G and F be graphs where G is 3-connected and has a t least 5 vertices and F has no isolated vertices; let e : EG + EF be a 1-1 map such that A E S3(G) e ( A ) E S3(F). Then there exists an isomorphism of G onto F inducing e, and this isomorphism is unique.
Let S',(G)denote the set of n-skeins of G with the distance S r between the terminals. Then the following strengthened version of Theorem 1 is also true.
+
Theorem 3. Let G and F be graphs where G is (n 1)-uertex connected and F has no isolated vertices ; let e : EG + EF be a 1-1 map such that A E S;(G)e e ( A ) E S?,(F). Then an isomorphism of G onto F exists and is the only one which induces e. 327
328
A.K. Kelmans
Note added in proof. Three different proofs of Theorem 2 were given in [3-51. A.K. Kelmans [6] found an example of a 4-skein automorphism of a 4-connected graph which was not induced by an automorphism of the graph and proved that this example is unique among the 4-skein isomorphisms of 4-connected graphs. The results of the note have been reported to the Moscow Discrete Mathematics Seminar (Institute of Control Sciences, Moscow, October 1979), to the AllUnion Graph Theory Seminar (Odessa, U.S.S.R., September 1980) and to the All-Union Conference on Statistic and Discrete Analyses of Non-Number Information (Alma-Ata, U.S.S.R., September 1981) [7].
References [ I ] R. Halin and H.A. Jung, J. London Math. SOC.42 (1967) 254-256. [2] H. Whitney, Amer. J. Math. 55 (1933) 245-254. [3] R.L. Heminger. H.A. Jung and A.K. Kelmans, On 3-skein isomorphisms of graphs, Combinatorica 4 (1982) 373-376. 141 A.K. Kelmans, On 3-skeins in a 3-connected graph, submitted to Combinatorica in July 1981. [S] A.K. Kelmans, A short proof and a strengthening of the Whitney 2-isomorphism theorem on graphs, in print. [6] A.K. Kelmans, On homeomorphic embeddings of graphs with given properties, in print. [7] A.K. Kelmans, 3-skeins in a 3-connected graph, in: First All-Union Conference on Statistic and Discrete Analyses of Non-Number Information, Moscow-Alma-Ata, 1981 (in Russian).
EDGE MAPS AND ISOMORPHISMS OF GRAPHS
A.K. KELMANS Profsouznaya Str. 130, 117321 Moscow, U.S.S.R.
Let G be an undirected graph and EG denote the edge-set of G. The union of n 2 1 chains of G is called an n-skein [l] if they have the common end-points (the terminals of the skein) and no pair of them has an inner point in common. Let S, ( G )denote the set of n-skeins of G. A 1-1 map e :EG .--, EF is called an n-skein isomorphism of G onto F if A ES,(G) @ e(A)€S.(F). In [l] the following result is obtained. Theorem 1. Suppose that G and F are (n + 1)-vertex connected, n 2 2. Then G and F a r e isomorphic if and only if they are n-skein isomorphic, and any n-skein isomorphism of G onto F is induced by a n isomorphism of G onto E
The same assertion for n-skeins in n-connected graphs is wrong, as Whitney [2] showed for n = 2. Also, it is easy to see [l] that every permutation of the edge-set of K4 is a 3-skein automorphism of K4, but clearly not every such map is induced by an automorphism of K,. In [l] we read: ". . . we were not able to find more complicated examples in the case n = 3, nor could we find for n 3 4 an example of an n-skein isomorphism of an n-vertex connected graph which was not induced by an isomorphism". Here we see that no other example of this kind exists for 3-connected graphs. Theorem 2. Let G and F be graphs where G is 3-connected and has a t least 5 vertices and F has no isolated vertices; let e : EG + EF be a 1-1 map such that A E S3(G) e ( A ) E S3(F). Then there exists an isomorphism of G onto F inducing e, and this isomorphism is unique.
Let S',(G)denote the set of n-skeins of G with the distance S r between the terminals. Then the following strengthened version of Theorem 1 is also true.
+
Theorem 3. Let G and F be graphs where G is (n 1)-uertex connected and F has no isolated vertices ; let e : EG + EF be a 1-1 map such that A E S;(G)e e ( A ) E S?,(F). Then an isomorphism of G onto F exists and is the only one which induces e. 327
328
A.K. Kelmans
Note added in proof. Three different proofs of Theorem 2 were given in [3-51. A.K. Kelmans [6] found an example of a 4-skein automorphism of a 4-connected graph which was not induced by an automorphism of the graph and proved that this example is unique among the 4-skein isomorphisms of 4-connected graphs. The results of the note have been reported to the Moscow Discrete Mathematics Seminar (Institute of Control Sciences, Moscow, October 1979), to the AllUnion Graph Theory Seminar (Odessa, U.S.S.R., September 1980) and to the All-Union Conference on Statistic and Discrete Analyses of Non-Number Information (Alma-Ata, U.S.S.R., September 1981) [7].
References [ I ] R. Halin and H.A. Jung, J. London Math. SOC.42 (1967) 254-256. [2] H. Whitney, Amer. J. Math. 55 (1933) 245-254. [3] R.L. Heminger. H.A. Jung and A.K. Kelmans, On 3-skein isomorphisms of graphs, Combinatorica 4 (1982) 373-376. 141 A.K. Kelmans, On 3-skeins in a 3-connected graph, submitted to Combinatorica in July 1981. [S] A.K. Kelmans, A short proof and a strengthening of the Whitney 2-isomorphism theorem on graphs, in print. [6] A.K. Kelmans, On homeomorphic embeddings of graphs with given properties, in print. [7] A.K. Kelmans, 3-skeins in a 3-connected graph, in: First All-Union Conference on Statistic and Discrete Analyses of Non-Number Information, Moscow-Alma-Ata, 1981 (in Russian).