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On Graphic-Minimal Spaces
Annals of Discrete Mathematics 8 (1980) 123-126 @ North-Holland Publishing Company
ON GRAPHIC-MINIMAL SPACES FranGois JAEGER Labomtoire I.M.A.G., B.P. 53 X , 38041 Grenoble Cedex, France
1. Introduction 1.1. Spaces Let E be a finite non-empty set; 9 ( E ) (the set of all subsets of E ) is considered as a vector space over GF(2) (the addition is the symmetric difference of sets); if 5$ c B(E)we shall denote by (%) the subspace of 9 ( E ) generated by the elements of 8 ;if 9 is a subspace of P ( E ) , the subspace of B(E) orthogonal to 9 is: 9l= {A E 9 ( E ) 1 V F E 9, \A n F (= 0 (mod 2)); the support of 9 is the subset ~ ( =9U) FFES of E. A space is any pair ( E , 9 ) where E is a finite non-empty set and 9 is a subspace of B(E) with a ( 9 ) = E ; two spaces ( E, 9)and (E‘,9’) are called isomorphic if there exists a bijection rp :E + E’ such that {cp(F)1 F ES}= 9;in this case we shall write (E, 9) = (E’,9’). Clearly = is an equivalence relation.
1.2. Spaces and graphs A graph G is a pair ( V ( G ) , E ( G ) )where V ( G ) is a finite non-empty set of vertices, E ( G ) is a finite non-empty set of edges, and to each edge corresponds an unordered pair of vertices called its ends. For every S E V ( G )let w G ( S ) be the set of edges of G with exactly one end in S; let X ( G )= (0 E E ( G ) 1 3s E V ( G ): 0 = o G ( S ) } ; X ( G ) is a subspace of 9 ( E ( G ) )and u ( X ( G ) )is the set of edges of G which are not loops (a loop is an edge with two identical ends); ( u ( X ( G ) ) X , ( G ) )is the cocycle space of G and will be denoted for short by X ( G ) . Let % ( G = ) [ X ( G ) ] l ;a ( % ( G )is ) the set of edges of G which are not bridges ( e E E ( G ) is a bridge if { e } E X ( G ) ) ;( a ( % ( G ) )% , ( G ) )is the cycle space of G and will be denoted for short by % ( G ) . Other definitions on graphs will be found in [l]. A space will be said to be cographic (respectively: graphic) if it is isomorphic to the cocycle space (respectively: cycle space) of some graph. A space will be said to be planar if it is both graphic and cographic. 123
F. Jaeger
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1.3. Series-extension and series-reduction Let ( E , 9)be a space; we shall say that the space (E’,9’)is a series-extension of ( E , 9 ) if there exists a mapping cp from E’ onto E such that 9’= {F’c E’1 ~ F 9Ewith F’ = CesFcp-’(e)}; equivalently we shall say that (E, 9‘is)a series-reduction of (E‘,9‘). 1.4. A preorder relation for spaces Let (E, 9)be a space; a covering subspace of (E, 9)is any space of the form (E, 9’)where 9’ is a subspace of 9 (note that we must have ~(9’) = o(9)= E ) . Let (E, 9) and (E’,sl) be two spaces. We shall write: (E, 9) <(E’, 9’)iff (E,9)is a series-reduction of some covering subspace of (E’,F). It is clear that < is a preorder relation; the associated equivalence relation is ; if (E, 9) s ( E ’ ,9’)and (E, 9)# (E’,9) we shall write (E, 9)< (E’,9’). Let % be a class of spaces; a space (E, 9) will be said to be %-minimal if it belongs to % and no space ( E’ ,9’)with (E’,9’)< (E, $) belongs to %. For every space ( E , 9 ) in % there exists a %-minimal space (E’,9’) with (E’,9’) (E, 9). A space will be said to be minimal (respectively: cographic-minimal, graphicminimal, planar-minimal) if it is %-minimal, where % is the class of all spaces (respectively: of cographic, graphic, planar spaces). 2-
2. Minimal spaces and the critical number For any integer n a l let E, = 9 ( { 1 , . . . , n } ) - { g } ; V i ~ ( 1 , ... , n } let x, = {A E E,, 1 i E A}; let X,, be the space (En,({xi 1 i = 1 , . . . , n})).
Proposition 1. A space is minimal i f it is isomorphic to some X , ( n 3 1 ) . The critical number of the space (E, 9) is the smallest integer C 3 1 such that E is the union of C elements of 9 (see [ 2 , 5 ] ) .
Proposition 2. The critical number of the space (E,9) is the smallest integer n 2 1 such that X,, s ( E , 9). 3. cogrrrphic-
*
-
1 spaces and the
chromatic number
Let C, be the complete graph (with no loops or multiple edges) on n vertices (n3 2).
Proposition 3. A space is cographic-minimal if it is isomorphic to some X(C,,) ( n 2 2 ,n f 4 ) .
On graphic-minimal spaces
125
Proposition 4. Let G be a loopZess graph, let y ( G ) be its chromatic number and let v ( G ) be the smallest integer n ( n a 2 , n f 4 ) such thatX(C,,)SX(G). Then v ( G ) S y ( G )< 2{"'Jb("(G))}.
4. Planar-minimal spaces and the Four-Cdor Theorem
Proposition 5. A space is planar-minimal i f f it is isomorphic to X1 or X,. Remark. This is an equivalent formulation of the Four-Color Theorem; the FourColor-Theorem can be stated as follows: (F-C-T) Every planar space has critical number at most 2. By Proposition 2 this is equivalent to: For every planar space (E, 9): X , G ( E ,5F) or X , S ( E , 9). Since X, and X , are planar (XI= X(C,); X , = X ( C J ) this is equivalent to Proposition 5.
5. Graphic-minimal spaces and a conjecture of Fulkerson Proposition 6. X , , X , and % ( P ) , where P is the Petersen graph, are graphicminimal spaces; every graphic-minimal space which is not isomorphic to one of these is isomorphic to some % ( G ) ,where G is a loopless cubic 3-edge-connected graph which can not be edge-colored with 3 colors and such that % ( P ) $ % ( G ) . Conjecture 1. Every graphic-minimal space is isomorphic to XI, X , or % ( P ) . A snark is a loopless cyclically 4-edge-connected cubic graph which cannot be edge-colored with 3 colors. Conjecture 1 is equivalent to:
Conjecture 1'. For every mark G, % ( P ) s % ( G ) . In [3], Fulkerson has proposed the following conjecture: Conjecture 2. For every bridgeless cubic graph G, by replacing every edge of G by two parallel edges one obtains a 6-regular graph which is edge-6-colorable. Let z e ( P ) be the space of cycles of even cardinality of the Petersen graph, i.e. %e(P)= ( E ( P ) , n (E(P))I).
Proposition 7. Conjecture 2 is equivalent to: Conjecture 2'. For every graphic space %, XI S % or X , S % or %,(P)G %.
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Since %,(P)s%(P), it follows that Conjecture 1 implies Conjecture 2 . Since X 3 s % ( P ) , Conjecture 1 also implies the following result proved in [4]: Proposition 8. Every graphic space has critical number at most 3 .
References [l] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1970). [2] H.H. Craw and G.C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries (M.I.T. Press, Cambridge, MA. 1970). [3] D.R. Fulkerson, Blocking and antiblocking pairs of polyhedra, Math. Programming 1 (1971) 168-194. [4] F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory 26 (B) (2) (1979) 205-216. [5] D.J.A. Welsh, Matroid Theory (Academic Press, London, 1976).
ON GRAPHIC-MINIMAL SPACES FranGois JAEGER Labomtoire I.M.A.G., B.P. 53 X , 38041 Grenoble Cedex, France
1. Introduction 1.1. Spaces Let E be a finite non-empty set; 9 ( E ) (the set of all subsets of E ) is considered as a vector space over GF(2) (the addition is the symmetric difference of sets); if 5$ c B(E)we shall denote by (%) the subspace of 9 ( E ) generated by the elements of 8 ;if 9 is a subspace of P ( E ) , the subspace of B(E) orthogonal to 9 is: 9l= {A E 9 ( E ) 1 V F E 9, \A n F (= 0 (mod 2)); the support of 9 is the subset ~ ( =9U) FFES of E. A space is any pair ( E , 9 ) where E is a finite non-empty set and 9 is a subspace of B(E) with a ( 9 ) = E ; two spaces ( E, 9)and (E‘,9’) are called isomorphic if there exists a bijection rp :E + E’ such that {cp(F)1 F ES}= 9;in this case we shall write (E, 9) = (E’,9’). Clearly = is an equivalence relation.
1.2. Spaces and graphs A graph G is a pair ( V ( G ) , E ( G ) )where V ( G ) is a finite non-empty set of vertices, E ( G ) is a finite non-empty set of edges, and to each edge corresponds an unordered pair of vertices called its ends. For every S E V ( G )let w G ( S ) be the set of edges of G with exactly one end in S; let X ( G )= (0 E E ( G ) 1 3s E V ( G ): 0 = o G ( S ) } ; X ( G ) is a subspace of 9 ( E ( G ) )and u ( X ( G ) )is the set of edges of G which are not loops (a loop is an edge with two identical ends); ( u ( X ( G ) ) X , ( G ) )is the cocycle space of G and will be denoted for short by X ( G ) . Let % ( G = ) [ X ( G ) ] l ;a ( % ( G )is ) the set of edges of G which are not bridges ( e E E ( G ) is a bridge if { e } E X ( G ) ) ;( a ( % ( G ) )% , ( G ) )is the cycle space of G and will be denoted for short by % ( G ) . Other definitions on graphs will be found in [l]. A space will be said to be cographic (respectively: graphic) if it is isomorphic to the cocycle space (respectively: cycle space) of some graph. A space will be said to be planar if it is both graphic and cographic. 123
F. Jaeger
124
1.3. Series-extension and series-reduction Let ( E , 9)be a space; we shall say that the space (E’,9’)is a series-extension of ( E , 9 ) if there exists a mapping cp from E’ onto E such that 9’= {F’c E’1 ~ F 9Ewith F’ = CesFcp-’(e)}; equivalently we shall say that (E, 9‘is)a series-reduction of (E‘,9‘). 1.4. A preorder relation for spaces Let (E, 9)be a space; a covering subspace of (E, 9)is any space of the form (E, 9’)where 9’ is a subspace of 9 (note that we must have ~(9’) = o(9)= E ) . Let (E, 9) and (E’,sl) be two spaces. We shall write: (E, 9) <(E’, 9’)iff (E,9)is a series-reduction of some covering subspace of (E’,F). It is clear that < is a preorder relation; the associated equivalence relation is ; if (E, 9) s ( E ’ ,9’)and (E, 9)# (E’,9) we shall write (E, 9)< (E’,9’). Let % be a class of spaces; a space (E, 9) will be said to be %-minimal if it belongs to % and no space ( E’ ,9’)with (E’,9’)< (E, $) belongs to %. For every space ( E , 9 ) in % there exists a %-minimal space (E’,9’) with (E’,9’) (E, 9). A space will be said to be minimal (respectively: cographic-minimal, graphicminimal, planar-minimal) if it is %-minimal, where % is the class of all spaces (respectively: of cographic, graphic, planar spaces). 2-
2. Minimal spaces and the critical number For any integer n a l let E, = 9 ( { 1 , . . . , n } ) - { g } ; V i ~ ( 1 , ... , n } let x, = {A E E,, 1 i E A}; let X,, be the space (En,({xi 1 i = 1 , . . . , n})).
Proposition 1. A space is minimal i f it is isomorphic to some X , ( n 3 1 ) . The critical number of the space (E, 9) is the smallest integer C 3 1 such that E is the union of C elements of 9 (see [ 2 , 5 ] ) .
Proposition 2. The critical number of the space (E,9) is the smallest integer n 2 1 such that X,, s ( E , 9). 3. cogrrrphic-
*
-
1 spaces and the
chromatic number
Let C, be the complete graph (with no loops or multiple edges) on n vertices (n3 2).
Proposition 3. A space is cographic-minimal if it is isomorphic to some X(C,,) ( n 2 2 ,n f 4 ) .
On graphic-minimal spaces
125
Proposition 4. Let G be a loopZess graph, let y ( G ) be its chromatic number and let v ( G ) be the smallest integer n ( n a 2 , n f 4 ) such thatX(C,,)SX(G). Then v ( G ) S y ( G )< 2{"'Jb("(G))}.
4. Planar-minimal spaces and the Four-Cdor Theorem
Proposition 5. A space is planar-minimal i f f it is isomorphic to X1 or X,. Remark. This is an equivalent formulation of the Four-Color Theorem; the FourColor-Theorem can be stated as follows: (F-C-T) Every planar space has critical number at most 2. By Proposition 2 this is equivalent to: For every planar space (E, 9): X , G ( E ,5F) or X , S ( E , 9). Since X, and X , are planar (XI= X(C,); X , = X ( C J ) this is equivalent to Proposition 5.
5. Graphic-minimal spaces and a conjecture of Fulkerson Proposition 6. X , , X , and % ( P ) , where P is the Petersen graph, are graphicminimal spaces; every graphic-minimal space which is not isomorphic to one of these is isomorphic to some % ( G ) ,where G is a loopless cubic 3-edge-connected graph which can not be edge-colored with 3 colors and such that % ( P ) $ % ( G ) . Conjecture 1. Every graphic-minimal space is isomorphic to XI, X , or % ( P ) . A snark is a loopless cyclically 4-edge-connected cubic graph which cannot be edge-colored with 3 colors. Conjecture 1 is equivalent to:
Conjecture 1'. For every mark G, % ( P ) s % ( G ) . In [3], Fulkerson has proposed the following conjecture: Conjecture 2. For every bridgeless cubic graph G, by replacing every edge of G by two parallel edges one obtains a 6-regular graph which is edge-6-colorable. Let z e ( P ) be the space of cycles of even cardinality of the Petersen graph, i.e. %e(P)= ( E ( P ) , n (E(P))I).
Proposition 7. Conjecture 2 is equivalent to: Conjecture 2'. For every graphic space %, XI S % or X , S % or %,(P)G %.
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Since %,(P)s%(P), it follows that Conjecture 1 implies Conjecture 2 . Since X 3 s % ( P ) , Conjecture 1 also implies the following result proved in [4]: Proposition 8. Every graphic space has critical number at most 3 .
References [l] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1970). [2] H.H. Craw and G.C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries (M.I.T. Press, Cambridge, MA. 1970). [3] D.R. Fulkerson, Blocking and antiblocking pairs of polyhedra, Math. Programming 1 (1971) 168-194. [4] F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory 26 (B) (2) (1979) 205-216. [5] D.J.A. Welsh, Matroid Theory (Academic Press, London, 1976).