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A case of hadwiger's conjecture
© DISCRETE MATHEMATICS 4 (1973) 1 9 7 - 1 9 9 . North-Holland Publishing Co m p a n y
A CASE OF HADWIGER'S CONJECTURE Michael O. ALBERTSON * Department of Mathematicsl University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Received 30 May 1 9 7 2 " *
Abstract. Sufficient conditions are established for Hadwiger's Conjecture to be valid in the case r = 5.
In this paper, we are concerned with vertex colorings, i.e., assignments of integers to the vertices of a graph. We recall Hadwiger's Conjecture. Hadwiger's Conjecture. I f G is a graph w i t h c h r o m a t i c n u m b e r r t h e n G is contractible to K r. Hadwiger's Conjecture is known to be true for r less than five. In the case o f r = 5, it is equivalent to the Four Color Conjecture. Theorem. S u p p o s e G is a graph w i t h c h r o m a t i c n u m b e r f i v e a n d critical vertex V 5 . F u r t h e r s u p p o s e that there is a f o u r coloring o f G - V 5 w i t h the p r o p e r t y that the color 4 is assigned to e x a c t l y o n e v e r t e x a m o n g those vertices o f G that have distance f r o m V 5 less than or equal to two. Then G is c o n t r a c t i b l e to K 5 . The remainder of the paper shall be devoted to proving this theorem. We fix a five coloring of G by extending the hypothesized four coloring of G - V 5 by coloring V5 the color 5. We note that the color 4 must be used to color exactly one vertex adjacent to V s and no vertex of distance two from Vs . If not, it is clear that G could be four-colored by * A u t h o r ' s current address: Swarthmore College, Swarthmore, Pa, 19801, USA. ** Original version received 11 January 1'972.
198
M.O. Albertson, A case o f Hadwiger's conjecture
assigning V 5 the c o l o r 4. Assume t h a t V4 is the v e r t e x adjacent to V 5 that is colored 4. With the fixed five coloring o f G for i = 1, 2, 3, we define C i to be the set o f vertices in G t h a t are assigned the c o l o r i and are adjacent to b o t h [/'4 and V s . L e m m a . C i ~ 0 f o r i = 1 , 2 , 3. P r o o f o f the L e m m a . S u p p o s e C 1 = 0- I n t e r c h a n g e colors in the c o m p o n e n t o f the (1,5) subgraph o f G c o n t a i n i n g V 5 . We claim t h a t no v e r t e x in G that is n o w c o l o r e d 5 is adjacent to any v e r t e x in G c o l o r e d 4. Suppose t h a t u is a v e r t e x in G t h a t is n o w c o l o r e d 5. I f u is adjacent to V4 , t h e n it m u s t have originally b e e n c o l o r e d 1 and adjacent to b o t h V4 and V 5 thus an e l e m e n t o f C 1 . However, u c a n n o t be adjacent to a n y v e r t e x o t h e r t h a n V4 t h a t is c o l o r e d 4. I f it were, t h e n the v e r t e x t h a t it is adjacent to t h a t which is c o l o r e d 4 m u s t be o f distance t w o f r o m V 5 and this violates the h y p o t h e s i s o f the t h e o r e m . Thus, since n o v e r t e x in G t h a t is n o w c o l o r e d 5 is adjacent to any v e r t e x c o l o r e d 4, we can f o u r - c o l o r G b y interchanging colors in the c o m p o n e n t s o f the (4,5) subgraph o f G t h a t c o n t a i n vertices colored 5. We n o w c o n t i n u e with the p r o o f o f the t h e o r e m . Pick a v e r t e x f r o m C 1 , say V 1 . With the fixed five coloring and v e r t e x V 1 we define Bi, i = 2, 3, to be the set o f all vertices in G that are c o l o r e d i and in the same c o m p o n e n t o f the (1, i) subgraph o f G c o n t a i n i n g V 1 . Case 1. E i t h e r B 2 A C 2 = ~ o r B 3 (3 C 3 = ~). S u p p o s e B 2 N C 2 = g). I n t e r c h a n g e colors in the c o m p o n e n t o f the (1,2) subgraph o f G containing V 1 . Since every v e r t e x in B 2 c a n n o t be adjacent to b o t h V4 and I15, we have a n e w coloring o f G that satisfies the initial h y p o t h e s i s o f the t h e o r e m and f e w e r e l e m e n t s in C 1 . We can c o n t i n u e this g a m b i t until e i t h e r C 1 = ~ w h i c h implies G is really f o u r colorable o r until B 2 nC 2:~oandB 3 nC 3 ~0. Case 2. S u p p o s e V 2 is i n B 2 n C 2 and V 3 is i n B 3 n C 3. See Fig. 1. If there is a (2,3) chain c o n n e c t i n g V 2 with V3, t h e n G is clearly contractible to K 5 . If V 2 and V 3 are n o t in the same c o m p o n e n t o f the (2,3) subgraph, i n t e r c h a n g e colors in the c o m p o n e n t o f the (2,3) subgraph o f G t h a t contains V 3 . We claim, either this decreases t h e cardinal-
M.O. Albertson, A case o f Hadwiger's conjecture
v,
2
199
v5
(2,3) path
3
Fig. 1.
ity o f B 3 n C3, or G is c o n t r a c t i b l e t o K 5 . S u p p o s e the v e r t e x u is in the B 3 A C 3 a f t e r the a b o v e i n t e r c h a n g e b u t was n o t before. T h e n u m u s t be a d j a c e n t to V 4 a n d V 5 and m u s t be c o n n e c t e d to V 3 b y a (2,3) path. F u r t h e r m o r e , a f t e r t h e i n t e r c h a n g e , u is c o n n e c t e d w i t h V 1 b y a (1,3) p a t h . This p a t h c a n n o t be i n c i d e n t w i t h V 3 . H e r e w i t h , o n e o b t a i n s t h r e e p a t h s being c o n t r a c t i b l e to a K 3 w i t h t h e vertices u, V 1 a n d V 3 , n a m e l y , the a b o v e (1,3) and (2,3) p a t h s ( t h e first p a t h t a k e n f r o m V 1 to its first c u t p o i n t w i t h the s e c o n d p a t h ) t o g e t h e r w i t h t h e (1,3) p a t h c o n n e c t i n g V 1 w i t h V 3 b e f o r e the i n t e r c h a n g e . H e n c e G is c o n t r a c t i b l e to K 5 . Since either G is c o n t r a c t i b l e to K 5 or the c a r d i n a l i t y o f B 3 n C 3 is r e d u c e d , we p r o c e e d as follows. A s s u m i n g G is n o t c o n t r a c t ible to K 5 , we can f o r c e B 3 n C 3 = 0 w h i c h p u t s us b a c k i n t o Case 1, w h e r e e i t h e r G is c o n t r a c t i b e t o K 5 or the c a r d i n a l i t y o f C 1 is r e d u c e d . H e n c e G m u s t be c o n t r a c t i b l e to K 5 .
References [ 1 ] G.A. Dirac, A theorem of R.L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. Soc. 73 (1957) 161-195. [2] H. Hadwiger, 0ber eine Klassifikation der Streckenkomplexe, Vierteljahresschr. Naturforsch. Ges. Z~irich 88 (1943) 133-142. [3] K. Wagner, Bemerkungen zum Vierfarbenproblem, Jber. Deutsch. Math.-Verein. 46 (1936) 26-32. [4] K. Wagner, Bemerkungen zu Hadwigers Vermutung, Math. Ann. 141 (1960) 4 3 3 - 4 5 1 .
A CASE OF HADWIGER'S CONJECTURE Michael O. ALBERTSON * Department of Mathematicsl University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Received 30 May 1 9 7 2 " *
Abstract. Sufficient conditions are established for Hadwiger's Conjecture to be valid in the case r = 5.
In this paper, we are concerned with vertex colorings, i.e., assignments of integers to the vertices of a graph. We recall Hadwiger's Conjecture. Hadwiger's Conjecture. I f G is a graph w i t h c h r o m a t i c n u m b e r r t h e n G is contractible to K r. Hadwiger's Conjecture is known to be true for r less than five. In the case o f r = 5, it is equivalent to the Four Color Conjecture. Theorem. S u p p o s e G is a graph w i t h c h r o m a t i c n u m b e r f i v e a n d critical vertex V 5 . F u r t h e r s u p p o s e that there is a f o u r coloring o f G - V 5 w i t h the p r o p e r t y that the color 4 is assigned to e x a c t l y o n e v e r t e x a m o n g those vertices o f G that have distance f r o m V 5 less than or equal to two. Then G is c o n t r a c t i b l e to K 5 . The remainder of the paper shall be devoted to proving this theorem. We fix a five coloring of G by extending the hypothesized four coloring of G - V 5 by coloring V5 the color 5. We note that the color 4 must be used to color exactly one vertex adjacent to V s and no vertex of distance two from Vs . If not, it is clear that G could be four-colored by * A u t h o r ' s current address: Swarthmore College, Swarthmore, Pa, 19801, USA. ** Original version received 11 January 1'972.
198
M.O. Albertson, A case o f Hadwiger's conjecture
assigning V 5 the c o l o r 4. Assume t h a t V4 is the v e r t e x adjacent to V 5 that is colored 4. With the fixed five coloring o f G for i = 1, 2, 3, we define C i to be the set o f vertices in G t h a t are assigned the c o l o r i and are adjacent to b o t h [/'4 and V s . L e m m a . C i ~ 0 f o r i = 1 , 2 , 3. P r o o f o f the L e m m a . S u p p o s e C 1 = 0- I n t e r c h a n g e colors in the c o m p o n e n t o f the (1,5) subgraph o f G c o n t a i n i n g V 5 . We claim t h a t no v e r t e x in G that is n o w c o l o r e d 5 is adjacent to any v e r t e x in G c o l o r e d 4. Suppose t h a t u is a v e r t e x in G t h a t is n o w c o l o r e d 5. I f u is adjacent to V4 , t h e n it m u s t have originally b e e n c o l o r e d 1 and adjacent to b o t h V4 and V 5 thus an e l e m e n t o f C 1 . However, u c a n n o t be adjacent to a n y v e r t e x o t h e r t h a n V4 t h a t is c o l o r e d 4. I f it were, t h e n the v e r t e x t h a t it is adjacent to t h a t which is c o l o r e d 4 m u s t be o f distance t w o f r o m V 5 and this violates the h y p o t h e s i s o f the t h e o r e m . Thus, since n o v e r t e x in G t h a t is n o w c o l o r e d 5 is adjacent to any v e r t e x c o l o r e d 4, we can f o u r - c o l o r G b y interchanging colors in the c o m p o n e n t s o f the (4,5) subgraph o f G t h a t c o n t a i n vertices colored 5. We n o w c o n t i n u e with the p r o o f o f the t h e o r e m . Pick a v e r t e x f r o m C 1 , say V 1 . With the fixed five coloring and v e r t e x V 1 we define Bi, i = 2, 3, to be the set o f all vertices in G that are c o l o r e d i and in the same c o m p o n e n t o f the (1, i) subgraph o f G c o n t a i n i n g V 1 . Case 1. E i t h e r B 2 A C 2 = ~ o r B 3 (3 C 3 = ~). S u p p o s e B 2 N C 2 = g). I n t e r c h a n g e colors in the c o m p o n e n t o f the (1,2) subgraph o f G containing V 1 . Since every v e r t e x in B 2 c a n n o t be adjacent to b o t h V4 and I15, we have a n e w coloring o f G that satisfies the initial h y p o t h e s i s o f the t h e o r e m and f e w e r e l e m e n t s in C 1 . We can c o n t i n u e this g a m b i t until e i t h e r C 1 = ~ w h i c h implies G is really f o u r colorable o r until B 2 nC 2:~oandB 3 nC 3 ~0. Case 2. S u p p o s e V 2 is i n B 2 n C 2 and V 3 is i n B 3 n C 3. See Fig. 1. If there is a (2,3) chain c o n n e c t i n g V 2 with V3, t h e n G is clearly contractible to K 5 . If V 2 and V 3 are n o t in the same c o m p o n e n t o f the (2,3) subgraph, i n t e r c h a n g e colors in the c o m p o n e n t o f the (2,3) subgraph o f G t h a t contains V 3 . We claim, either this decreases t h e cardinal-
M.O. Albertson, A case o f Hadwiger's conjecture
v,
2
199
v5
(2,3) path
3
Fig. 1.
ity o f B 3 n C3, or G is c o n t r a c t i b l e t o K 5 . S u p p o s e the v e r t e x u is in the B 3 A C 3 a f t e r the a b o v e i n t e r c h a n g e b u t was n o t before. T h e n u m u s t be a d j a c e n t to V 4 a n d V 5 and m u s t be c o n n e c t e d to V 3 b y a (2,3) path. F u r t h e r m o r e , a f t e r t h e i n t e r c h a n g e , u is c o n n e c t e d w i t h V 1 b y a (1,3) p a t h . This p a t h c a n n o t be i n c i d e n t w i t h V 3 . H e r e w i t h , o n e o b t a i n s t h r e e p a t h s being c o n t r a c t i b l e to a K 3 w i t h t h e vertices u, V 1 a n d V 3 , n a m e l y , the a b o v e (1,3) and (2,3) p a t h s ( t h e first p a t h t a k e n f r o m V 1 to its first c u t p o i n t w i t h the s e c o n d p a t h ) t o g e t h e r w i t h t h e (1,3) p a t h c o n n e c t i n g V 1 w i t h V 3 b e f o r e the i n t e r c h a n g e . H e n c e G is c o n t r a c t i b l e to K 5 . Since either G is c o n t r a c t i b l e to K 5 or the c a r d i n a l i t y o f B 3 n C 3 is r e d u c e d , we p r o c e e d as follows. A s s u m i n g G is n o t c o n t r a c t ible to K 5 , we can f o r c e B 3 n C 3 = 0 w h i c h p u t s us b a c k i n t o Case 1, w h e r e e i t h e r G is c o n t r a c t i b e t o K 5 or the c a r d i n a l i t y o f C 1 is r e d u c e d . H e n c e G m u s t be c o n t r a c t i b l e to K 5 .
References [ 1 ] G.A. Dirac, A theorem of R.L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. Soc. 73 (1957) 161-195. [2] H. Hadwiger, 0ber eine Klassifikation der Streckenkomplexe, Vierteljahresschr. Naturforsch. Ges. Z~irich 88 (1943) 133-142. [3] K. Wagner, Bemerkungen zum Vierfarbenproblem, Jber. Deutsch. Math.-Verein. 46 (1936) 26-32. [4] K. Wagner, Bemerkungen zu Hadwigers Vermutung, Math. Ann. 141 (1960) 4 3 3 - 4 5 1 .