On non-normal 4-valent arc transitive dihedrants Aleksander Malni c - PowerPoint PPT Presentation

On non-normal 4-valent arc transitive dihedrants Aleksander Malniˇ c University of Ljubljana Joint work with Istv´ an Kov´ acs and Boˇ stjan Kuzman Banff, Canada November, 2008 1 / 19

Dihedrants and Bicirculants An n-dihedrant is a Cayley graph of a dihedral group D n . An n-bicirculant is a regular Z n -cover of a dipole. n-dihedrant ⇒ n-bicirculant It is often convenient if we consider dihedrants as bicirculants. 2 / 19

Dihedrants and Bicirculants An n-dihedrant is a Cayley graph of a dihedral group D n . An n-bicirculant is a regular Z n -cover of a dipole. n-dihedrant ⇒ n-bicirculant It is often convenient if we consider dihedrants as bicirculants. BC n 1 , BC n 2 , BC n 3 , BC n 4 Bicirculants of valency 4 fall into 4 classes, wrt. the number of perfect matchings between the two orbits of Z n . 2 / 19

4 valent edge transitive bicirculants BC n 1, BC n 3 No such graphs. Kov´ acs, Kuzman, M., Wilson, 2008 3 / 19

4 valent edge transitive bicirculants BC n 1, BC n 3 No such graphs. Kov´ acs, Kuzman, M., Wilson, 2008 BC n 2 Rose window graphs. Kov´ acs, Kutnar, Maruˇ siˇ c, 2008 Some of these are dihedrans. Generalized Rose window graphs. Still open. Conjecture: empty. 3 / 19

4 valent edge transitive bicirculants BC n 1, BC n 3 No such graphs. Kov´ acs, Kuzman, M., Wilson, 2008 BC n 2 Rose window graphs. Kov´ acs, Kutnar, Maruˇ siˇ c, 2008 Some of these are dihedrans. Generalized Rose window graphs. Still open. Conjecture: empty. 3 / 19

4 valent edge transitive bicirculants BC n 1, BC n 3 No such graphs. Kov´ acs, Kuzman, M., Wilson, 2008 BC n 2 Rose window graphs. Kov´ acs, Kutnar, Maruˇ siˇ c, 2008 Some of these are dihedrans. Generalized Rose window graphs. Still open. Conjecture: empty. BC n 4 X ∈ BC n 4 is necessarily a dihedrant. Normal, D n ⊳ Aut ( X ). Kov´ acs, Kuzman, M., Wilson, 2008 Non-normal. Kov´ acs, Kuzman, M., 2008 This talk. 3 / 19

Non-normal arc-transitive BC n 4 II. The graph K 5 , 5 − 5 K 2 . I. The lexicographic product C n [2 K 1 ]. n = 5, S = { b, ba, ba 2 , ba 3 } . n 2 , ba n 2 +1 } n ≥ 4 even, S = { b, ba, ba (in picture, n = 16). III. The non-incidence graph of IV. The incidence graph of PG (2 , 3). PG (2 , 2). n = 13, S = { b, ba, ba 3 , ba 9 } . n = 7, S = { b, ba, ba 2 , ba 4 } . V. A 2-cover of the graph III. VI. A 3-cover of the graph II. n = 14, S = { b, ba, ba 4 , ba 6 } . n = 15, S = { b, ba, ba 3 , ba 7 } Table 1: Non-normal 4-valent arc-transitive dihedrants satisfying the bipartition condition. 4 / 19

References Arc transitive dihedrants val 4, 1-regular C. Q. Wang, M. Xu, Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D 2 n , European J. Combin. 27 (2006), 750–766. C. Q. Wang, Z. Y. Zhou, One-regularity of 4-valent and normal Cayley graphs of dihedral groups D 2 n , Acta Math. Sinica (Chin. ser.) 49 (2006), 669–678. Y. H. Kwak, Y. M. Oh, One-regular normal Cayley graphs on dihedral groups of valency 4 or 6 with cyclic vertex stabilizer, Acta Math. Sinica (Engl. ser.) 22 (2006), 1305–1320. 5 / 19

References Arc transitive dihedrants val 4, 1-regular C. Q. Wang, M. Xu, Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D 2 n , European J. Combin. 27 (2006), 750–766. C. Q. Wang, Z. Y. Zhou, One-regularity of 4-valent and normal Cayley graphs of dihedral groups D 2 n , Acta Math. Sinica (Chin. ser.) 49 (2006), 669–678. Y. H. Kwak, Y. M. Oh, One-regular normal Cayley graphs on dihedral groups of valency 4 or 6 with cyclic vertex stabilizer, Acta Math. Sinica (Engl. ser.) 22 (2006), 1305–1320. Lemma X ∈ BC n 4 non-normal ⇒ X not 1-regular. 5 / 19

References Arc transitive dihedrants val 4, 1-regular C. Q. Wang, M. Xu, Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D 2 n , European J. Combin. 27 (2006), 750–766. C. Q. Wang, Z. Y. Zhou, One-regularity of 4-valent and normal Cayley graphs of dihedral groups D 2 n , Acta Math. Sinica (Chin. ser.) 49 (2006), 669–678. Y. H. Kwak, Y. M. Oh, One-regular normal Cayley graphs on dihedral groups of valency 4 or 6 with cyclic vertex stabilizer, Acta Math. Sinica (Engl. ser.) 22 (2006), 1305–1320. Lemma X ∈ BC n 4 non-normal ⇒ X not 1-regular. 2-arc transitive dihedrants S. F. Du, M., D. Maruˇ siˇ c, Classification of 2-arc transitive dihedrants, J.Combin. Theory B , in print. 5 / 19

Non-normal arc-transitive BC n 4 II. The graph K 5 , 5 − 5 K 2 . I. The lexicographic product C n [2 K 1 ]. n = 5, S = { b, ba, ba 2 , ba 3 } . n 2 , ba n 2 +1 } n ≥ 4 even, S = { b, ba, ba (in picture, n = 16). III. The non-incidence graph of IV. The incidence graph of PG (2 , 3). PG (2 , 2). n = 13, S = { b, ba, ba 3 , ba 9 } . n = 7, S = { b, ba, ba 2 , ba 4 } . V. A 2-cover of the graph III. VI. A 3-cover of the graph II. n = 14, S = { b, ba, ba 4 , ba 6 } . n = 15, S = { b, ba, ba 3 , ba 7 } Table 1: Non-normal 4-valent arc-transitive dihedrants satisfying the bipartition condition. 6 / 19

Non-normal arc transitive BC n 4, Reduction to circulants With X ∈ BC n 4 we associate a certain circulant Y (step two blue graph in figure below). Graphs X are classified by finding all possible graphs Y . 7 / 19

Non-normal arc transitive BC n 4, Reduction to circulants With X ∈ BC n 4 we associate a certain circulant Y (step two blue graph in figure below). Graphs X are classified by finding all possible graphs Y . Y. G. Baik, Y. Q. Feng, H. S. Sim, M. Y. Xu, On the normality of Cayley graphs of abelian groups, Algebra Colloq. 5 (1998), 227–234. I. Kov´ acs, Classifying Arc-Transitive Circulants, J. Algebraic Combin. 20 (2004), 353–358. C. H. Li, Permutation groups with a cylic regular subgroup and arc-transitive circulants, J. Algebraic Combin. 21 (2005), 131-136. 7 / 19

Non-normal arc transitive BC n 4, Reduction to circulants With X ∈ BC n 4 we associate a certain circulant Y (step two blue graph in figure below). Graphs X are classified by finding all possible graphs Y . Y. G. Baik, Y. Q. Feng, H. S. Sim, M. Y. Xu, On the normality of Cayley graphs of abelian groups, Algebra Colloq. 5 (1998), 227–234. I. Kov´ acs, Classifying Arc-Transitive Circulants, J. Algebraic Combin. 20 (2004), 353–358. C. H. Li, Permutation groups with a cylic regular subgroup and arc-transitive circulants, J. Algebraic Combin. 21 (2005), 131-136. In order this to work we need to trasfer symmetry properties between X and Y 7 / 19

���� � � ������ �� � � ������ � Reduction to circulants, Why it works? Lemma X ∈ BC n 4 non-normal ⇔ X non-normal Z n -cover of dip 4 . 8 / 19

� ���� � ������ �� � � ������ � Reduction to circulants, Why it works? Lemma X ∈ BC n 4 non-normal ⇔ X non-normal Z n -cover of dip 4 . Lemma X ∈ BC n 4 non-normal ⇒ Y non-normal circulant. 8 / 19

� � � ������ ������ �� � ���� � Reduction to circulants, Why it works? Lemma X ∈ BC n 4 non-normal ⇔ X non-normal Z n -cover of dip 4 . Lemma X ∈ BC n 4 non-normal ⇒ Y non-normal circulant. Ker φ = Ker ¯ G = 1. Then Z ⊳ Aut ( Y ) ⇒ Z ⊳ ¯ G ⇒ Z ⊳ G Ker φ = Ker ¯ G � = 1. Then X = C n [2 K 1 ], n ≥ 4 even, Y = C n / 2 [ K 2 ]. 8 / 19

Why it works? The structure of Y X = Cay ( D n , S ) , S = { b , ba x , ba y , ba z } T = { a ± x , a ± y , a ± z , a ± ( x − y ) , a ± ( y − z ) , a ± ( x − z ) } Y = Cay ( Z n , T ) , might not be arc transitive. However: it is an edge-disjoint union of arc transitive circulants (of which at least one of them is connected). 9 / 19

Why it works? The structure of Y X = Cay ( D n , S ) , S = { b , ba x , ba y , ba z } T = { a ± x , a ± y , a ± z , a ± ( x − y ) , a ± ( y − z ) , a ± ( x − z ) } Y = Cay ( Z n , T ) , might not be arc transitive. However: it is an edge-disjoint union of arc transitive circulants (of which at least one of them is connected). Lemma Either Y connected arc transitive, non normal, and T = { a ± x , a ± y , a ± z , a ± ( x − y ) , a ± ( y − z ) , a ± ( x − z ) } or Y = Y 1 + Y 2 , where Y 2 is connected, arc transitive, non-normal, and T 1 = { a ± x , a ± ( y − z ) } , T 2 = { a ± y , a ± z , a ± ( x − y ) , a ± ( x − z ) } 9 / 19

The structure of Y , uniformity index of X For e ∈ Y , let r ( e ) be the number of 3-cycles in X ∪ X 2 containing e . 10 / 19

The structure of Y , uniformity index of X For e ∈ Y , let r ( e ) be the number of 3-cycles in X ∪ X 2 containing e . Lemma If Y is arc transitive, then r ( e ) = 12 / | T | for each e ∈ E ( Y ) . If Y = Y 1 + Y 2 is an edge disjoint union of two arc transitive graphs, then � 4 / | T 1 | , for each e ∈ E ( Y 1 ) r ( e ) = 8 / | T 2 | , for each e ∈ E ( Y 2 ) . 10 / 19

The structure of Y , uniformity index of X For e ∈ Y , let r ( e ) be the number of 3-cycles in X ∪ X 2 containing e . Lemma If Y is arc transitive, then r ( e ) = 12 / | T | for each e ∈ E ( Y ) . If Y = Y 1 + Y 2 is an edge disjoint union of two arc transitive graphs, then � 4 / | T 1 | , for each e ∈ E ( Y 1 ) r ( e ) = 8 / | T 2 | , for each e ∈ E ( Y 2 ) . X is k -uniform if r ( e ) = k for all e ∈ E ( Y ). The paramter k is the uniformity index . If Y = Y 1 + Y 2 , then Y 1 is k 1 -uniform and Y 2 is k 2 -uniform. Possibly, k 1 = k 2 = k . 10 / 19

X is non-uniform Then Y = Y 1 + Y 2 . Since | T 1 | k 1 = 4, | T 2 | k 2 = 8 ⇒ k 1 , k 2 ∈ { 1 , 2 , 4 } . 11 / 19