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Periodicity of graph operators
Discrete Mathematics 235 (2001) 349–351
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Periodicity of graph operators Bohdan Zelinka Department of Applied Mathematics, Voronez ska 13, 460001 Liberec 1, Czech Republic
This lecture treats some problems from the book in [2] by E. Prisner. All these problems concern graph operators. A graph operator is a mapping to which every graph G from some class of graphs assigns a new graph (G). Usually, the class C of all undirected graphs without loops with multiple edges or the class Cf of all .nite graphs from C is used. (Here Cf is used in the case of the operator Pow2 .) A typical example of a graph operator is the line graph operator L which to every graph G from C assigns its line graph L(G). If is a graph operator and k is a positive integer, then the operator k is de.ned recurrently so that 1 = and k (G) = (k−1 (G)) for every integer k¿2 and for every graph G. A graph G is said to be periodic with period k (shortly k-periodic) in operator , if k (G) ∼ = G and i (G) G for 0 ¡ i ¡ k. If G is k-periodic for k = 1 in , it is .xed in . One problem from [2] is whether there exists a graph which is k-periodic for some k = 1 in the line graph operator L. (Graphs .xed in L are e.g. all circuits.) Another problem asks for the same with 4 instead of L. The operator 4 is a particular case of the operator m . A subgraph of a graph G which is a complete graph with p vertices is called a p-simplex of G (or shortly simplex of G). If a simplex of G is not a proper subgraph of another simplex of G, it is a clique (p-clique) of G. If m is a positive integer, then m is a graph operator to which a graph G assigns the graph m (G) so that the vertices of m (G) are all m-simplices of G and all p-cliques of G for all p ¡ m and two vertices are adjacent in m (G) if and only if they have a non-empty intersection. Note that if m¿2 and G is without triangles and without isolated vertices, then m (G) ∼ = L(G). For the graph K1 consisting of one isolated vertex we have m (K1 ) = K1 , while L(K1 ) = K0 ; the graph K0 is the empty graph, i.e. V (K0 ) = E(K0 ) = ∅. We consider this graph as connected. The answer to the above-mentioned problem follows from a rather general theorem. If to the disjoint union of graphs G1 and G2 an operator assigns the disjoint union of the graphs (G1 ) and (G2 ), then is called linear. This is a simpli.cation of the terminology from [2]. Most of the commonly used graph operators have this property.
E-mail address: [email protected] (B. Zelinka). c 2001 Elsevier Science B.V. All rights reserved. 0012-365X/01/$ - see front matter PII: S 0 0 1 2 - 3 6 5 X ( 0 0 ) 0 0 2 8 8 - 0
350
B. Zelinka / Discrete Mathematics 235 (2001) 349–351
Theorem 1 (Zelinka [3]). Let be a linear operator on C. Let there exist a sequence (Hn )∞ n=1 of pairwise non-isomorphic connected graphs with the property that (H0 ) = H0 and (Hn ) = Hn−1 for each positive integer n. Let k be an arbitrary positive integer. Then there exists a graph G which is k-periodic in the operator . The required graph G can be obtained as the disjoint union of graphs Hn for all numbers n which are multiples of k and of in.nitely many pairwise disjoint copies of H0 . This theorem yields the aFrmative answer for = L and for = n . Let Pn denote the path of length n, i.e. with n edges. For = L we may put H0 = K0 ; Hn = Pn−1 for n¿1. For = n we may put Hn = Pn for all non-negative integers n. Another problem from [2] concerns the operator Simp. To every graph G it assigns the simplex graph Simp(G) whose vertex set is the set of all simplices in G and in which two vertices are adjacent if and only if they have a non-empty intersection. Note that also 1-simplices are considered. The problem asks whether there are periodic graphs in Simp, except graphs consisting of isolated vertices. Theorem 2 (Zelinka [4]). Let G be a graph having at least one edge and such that |V (G)| ¡ ℵ! ; where ! is the least in8nite ordinal number. Then G is not periodic in the operator Simp. Proof is in [4]. Havel and the author studied the problem concerning the periodicity of the operator Pow2 , which is the complement of the second power. For a graph G the graph Pow2 (G) is the second power of G, i.e. the graph G 2 such that V (G 2 ) = V (G) and {x; y} ∈ E(G 2 ) if and only if 16dG (x; y)62, where dG denotes the distance in G. The graph Pow2 (G) is the complement of Pow2 (G). The problem asks, whether there are k-periodic graphs in Pow2 for k = 1. (An example for k = 1 is the circuit C7 of length 7.) Let G be a connected bipartite graph; then, its bipartition classes are uniquely determined and we may consider a complete bipartite graph on them. The complement of G with respect to this complete bipartite graph will be denoted by CB(G). Therefore, if A; B are the bipartition classes of G, then V (CB(G)) = V (G) = A ∪ B and vertices x ∈ A and y ∈ B are adjacent in CB(G) if and only if they are not adjacent in G. Theorem 3. Let G be a bipartite graph of diameter 3 and such that the graph CB(G) has also diameter 3. Then Pow2 (G) = CB(G); Pow2 (Pow2 (G)) = G. Proof is in [1]. Note that the operator Pow2 does not change the vertex set and in this theorem we have the sign of equality, i.e. the considered graphs are not only isomorphic, but equal with the same vertex set and the same edge set. If CB(G) is not isomorphic to G, then G is 2-periodic in Pow2 . For .nding such graphs the following theorem serves.
B. Zelinka / Discrete Mathematics 235 (2001) 349–351
351
Theorem 4. A complete bipartite graph Km; n can be decomposed into two edge-disjoint subgraphs of diameter 3 if and only if m¿6 and n¿6. Examples of required graphs are graphs of all .nite Desarguesian projective geometries. In such a graph the vertex set is the union of the point set and the line set and a point and a line are adjacent if and only if they are incident in the geometry. Other examples are among generalized hypercubes. Let n be a positive integer, let S ⊆{1; : : : ; n}. The generalized hypercube Qn (S) is the graph whose vertex set is the set of all Boolean vectors of dimension n and in which two vertices are adjacent if and only if their Hamming distance is in S. If S consists of odd numbers, then the graph Qn (S) is bipartite. The simplest generalized hypercube satisfying the condition are Q5 ({1; 5}) and Q5 ({3}). We have Pow2 (Q5 ({1; 5})) = Q5 ({1; 2; 4; 5}); Pow2 (Q5 ({1; 5})) = Q5 ({3}); Pow2 (Pow2 (Q5 ({1; 5}))) = Q5 ({2; 3; 4}); Pow2 (Pow2 (Q5 ({1; 5}))) = Q5 ({1; 5}): References [1] [2] [3] [4]
I. Havel, B. Zelinka, On 2-periodic graphs of a certain graph operator, Discussiones Math., in refereeing. E. Prisner, Graph Dynamics, Longman House, Burnt Mill, Harlow, 1995. B. Zelinka, A remark on graph operators, Math. Bohemica 124 (1999) 83–85. B. Zelinka, On the simplex graph operator, Discussiones Math. Graph Theory 18 (1998) 165–169.
www.elsevier.com/locate/disc
Periodicity of graph operators Bohdan Zelinka Department of Applied Mathematics, Voronez ska 13, 460001 Liberec 1, Czech Republic
This lecture treats some problems from the book in [2] by E. Prisner. All these problems concern graph operators. A graph operator is a mapping to which every graph G from some class of graphs assigns a new graph (G). Usually, the class C of all undirected graphs without loops with multiple edges or the class Cf of all .nite graphs from C is used. (Here Cf is used in the case of the operator Pow2 .) A typical example of a graph operator is the line graph operator L which to every graph G from C assigns its line graph L(G). If is a graph operator and k is a positive integer, then the operator k is de.ned recurrently so that 1 = and k (G) = (k−1 (G)) for every integer k¿2 and for every graph G. A graph G is said to be periodic with period k (shortly k-periodic) in operator , if k (G) ∼ = G and i (G) G for 0 ¡ i ¡ k. If G is k-periodic for k = 1 in , it is .xed in . One problem from [2] is whether there exists a graph which is k-periodic for some k = 1 in the line graph operator L. (Graphs .xed in L are e.g. all circuits.) Another problem asks for the same with 4 instead of L. The operator 4 is a particular case of the operator m . A subgraph of a graph G which is a complete graph with p vertices is called a p-simplex of G (or shortly simplex of G). If a simplex of G is not a proper subgraph of another simplex of G, it is a clique (p-clique) of G. If m is a positive integer, then m is a graph operator to which a graph G assigns the graph m (G) so that the vertices of m (G) are all m-simplices of G and all p-cliques of G for all p ¡ m and two vertices are adjacent in m (G) if and only if they have a non-empty intersection. Note that if m¿2 and G is without triangles and without isolated vertices, then m (G) ∼ = L(G). For the graph K1 consisting of one isolated vertex we have m (K1 ) = K1 , while L(K1 ) = K0 ; the graph K0 is the empty graph, i.e. V (K0 ) = E(K0 ) = ∅. We consider this graph as connected. The answer to the above-mentioned problem follows from a rather general theorem. If to the disjoint union of graphs G1 and G2 an operator assigns the disjoint union of the graphs (G1 ) and (G2 ), then is called linear. This is a simpli.cation of the terminology from [2]. Most of the commonly used graph operators have this property.
E-mail address: [email protected] (B. Zelinka). c 2001 Elsevier Science B.V. All rights reserved. 0012-365X/01/$ - see front matter PII: S 0 0 1 2 - 3 6 5 X ( 0 0 ) 0 0 2 8 8 - 0
350
B. Zelinka / Discrete Mathematics 235 (2001) 349–351
Theorem 1 (Zelinka [3]). Let be a linear operator on C. Let there exist a sequence (Hn )∞ n=1 of pairwise non-isomorphic connected graphs with the property that (H0 ) = H0 and (Hn ) = Hn−1 for each positive integer n. Let k be an arbitrary positive integer. Then there exists a graph G which is k-periodic in the operator . The required graph G can be obtained as the disjoint union of graphs Hn for all numbers n which are multiples of k and of in.nitely many pairwise disjoint copies of H0 . This theorem yields the aFrmative answer for = L and for = n . Let Pn denote the path of length n, i.e. with n edges. For = L we may put H0 = K0 ; Hn = Pn−1 for n¿1. For = n we may put Hn = Pn for all non-negative integers n. Another problem from [2] concerns the operator Simp. To every graph G it assigns the simplex graph Simp(G) whose vertex set is the set of all simplices in G and in which two vertices are adjacent if and only if they have a non-empty intersection. Note that also 1-simplices are considered. The problem asks whether there are periodic graphs in Simp, except graphs consisting of isolated vertices. Theorem 2 (Zelinka [4]). Let G be a graph having at least one edge and such that |V (G)| ¡ ℵ! ; where ! is the least in8nite ordinal number. Then G is not periodic in the operator Simp. Proof is in [4]. Havel and the author studied the problem concerning the periodicity of the operator Pow2 , which is the complement of the second power. For a graph G the graph Pow2 (G) is the second power of G, i.e. the graph G 2 such that V (G 2 ) = V (G) and {x; y} ∈ E(G 2 ) if and only if 16dG (x; y)62, where dG denotes the distance in G. The graph Pow2 (G) is the complement of Pow2 (G). The problem asks, whether there are k-periodic graphs in Pow2 for k = 1. (An example for k = 1 is the circuit C7 of length 7.) Let G be a connected bipartite graph; then, its bipartition classes are uniquely determined and we may consider a complete bipartite graph on them. The complement of G with respect to this complete bipartite graph will be denoted by CB(G). Therefore, if A; B are the bipartition classes of G, then V (CB(G)) = V (G) = A ∪ B and vertices x ∈ A and y ∈ B are adjacent in CB(G) if and only if they are not adjacent in G. Theorem 3. Let G be a bipartite graph of diameter 3 and such that the graph CB(G) has also diameter 3. Then Pow2 (G) = CB(G); Pow2 (Pow2 (G)) = G. Proof is in [1]. Note that the operator Pow2 does not change the vertex set and in this theorem we have the sign of equality, i.e. the considered graphs are not only isomorphic, but equal with the same vertex set and the same edge set. If CB(G) is not isomorphic to G, then G is 2-periodic in Pow2 . For .nding such graphs the following theorem serves.
B. Zelinka / Discrete Mathematics 235 (2001) 349–351
351
Theorem 4. A complete bipartite graph Km; n can be decomposed into two edge-disjoint subgraphs of diameter 3 if and only if m¿6 and n¿6. Examples of required graphs are graphs of all .nite Desarguesian projective geometries. In such a graph the vertex set is the union of the point set and the line set and a point and a line are adjacent if and only if they are incident in the geometry. Other examples are among generalized hypercubes. Let n be a positive integer, let S ⊆{1; : : : ; n}. The generalized hypercube Qn (S) is the graph whose vertex set is the set of all Boolean vectors of dimension n and in which two vertices are adjacent if and only if their Hamming distance is in S. If S consists of odd numbers, then the graph Qn (S) is bipartite. The simplest generalized hypercube satisfying the condition are Q5 ({1; 5}) and Q5 ({3}). We have Pow2 (Q5 ({1; 5})) = Q5 ({1; 2; 4; 5}); Pow2 (Q5 ({1; 5})) = Q5 ({3}); Pow2 (Pow2 (Q5 ({1; 5}))) = Q5 ({2; 3; 4}); Pow2 (Pow2 (Q5 ({1; 5}))) = Q5 ({1; 5}): References [1] [2] [3] [4]
I. Havel, B. Zelinka, On 2-periodic graphs of a certain graph operator, Discussiones Math., in refereeing. E. Prisner, Graph Dynamics, Longman House, Burnt Mill, Harlow, 1995. B. Zelinka, A remark on graph operators, Math. Bohemica 124 (1999) 83–85. B. Zelinka, On the simplex graph operator, Discussiones Math. Graph Theory 18 (1998) 165–169.