ADVANCES IN MATHEMATICS(CHINA)

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1 Æ Ý ¹ ADVANCES IN MATHEMATICS(CHINA) 0 median Đ Ó ( ºÕ ³,, ÓÚ, ) doi: /sxjz b : u,v ¹ w G, z ÁÇÉ Ë½ ±, È z À u,v ¹ w Ä median. G À²Ï median, G Î Å Ì ÆÄ median. Ã ²Ï median µ» ÂÍ, ¾ ²Ï median ³, ²Ï median, ºÊ ²Ï median ¼ ²Ï median. Ì: ± ; median; ²Ï median ; ¼ MR(2000) Ý : 05C12; 05C75 / : O157.5 ÉØÐ: A u,v,w É G Â, Ì z ÁÖ ¾Â Ä¾, z Ô u,v,w median. Â G Ô median, G Â u,v,w» Ó Â median. Ñ median Á µ. Mulder ÖÍ median Û É ß, «¾ [1]. ÇÔ ÐÉ, median ¹ Î ØÆ ¼ É È ß [2 6]. median ß Þ Â Ç, 1947 Birkhoff Kiss [7] ±Í, Avann 1961 median Æ ÄÇ [8]. É Ì ¾ Ô median É Nebeský [9]. È median Â Ñ º [10] [1,11]. Chung, Graham Saks median À À «Å Á ß, «À ² ß ( Median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature ) [12]. ¼ median ØÆ, ¹ Å, «ß ØÆ. quasi-median [1] semi-median [13] almost median [13] weakly median [14] µ. À, Â Â median É ØÆ±? Ð Â ±? Ô, median, Ù Ã ¹ : Â G Ô median, Ì G Â» Â median. Þ Â median»é median,. Ñ Æ K 2,3 É median, É median, median É median Â. Ò ß median Æ ². 1 È Ù G = (V,E) É À, ¹ V Æ G, E Æ G ËÀ : ÀË² : Ù : Ë Á Ù (No , No ). wgfmath@126.com

2 2 Ü. ÁÔ, Å V(G) E(G) Æ G. Â «p = v 0 e 1 v 1 e 2...v k 1 e k v k ĐÔ Ô, 1 i k, e i ½ Ô v i 1 v i, ¾¹ Đ, p Ô v 0 ± v k, Ð Ô v 0 v k. k Ô p ¼. v 0 v k ¼Ä Ô v 0 v k Î, Ä¾ ¼ Ô v 0 v k, Ô d G (v 0,v k ). G Ü, Ð Ô d(v 0,v k ). p ¼ v 0 v k, Á p ÔÖ, C k Æ, ¹ k Ô ¼. k Ô», C k ÔÔÖ, ÔÒÖ. G V ³ Â V 1 V 2, Ä³ G Â½ V 1, Â½ V 2, G Ô ÊÐ. Û, V 1 Â V 2 Â» Ð, G Ô ÊÐ. V 1 = m, V 2 = n, ¾ Ñ Æ Ô K m,n. ¹ Õ¹ Ï± Ï Ñ [15]. Í«1 G = (V,E) u,v,w V(G). G z Ô u,v w Â median, Ì ÁÖ Ä¾ u v Ä¾ u w Ä¾ v w. Í«2 G u v Ô½ Õ ¹ Ô Ö u v Ä¾ Æ, Ô I G (u,v), I G (u,v) = {w V(G) d G (u,w)+d G (w,v) = d G (u,v)}, Ð Ô I(u,v). ¾, u,v w median É I(u,v) I(u,w) I(v,w). Í«3 G Ô median Ð, Ì G Â u,v,w,»½ Â median, É I(u,v) I(u,w) I(v,w) = 1. Í«4 G ÔÏ median Ð, Ì G Â u,v w,» Â median, É I(u,v) I(u,w) I(v,w) 1. Þ Â median»é median,. Ñ Æ K m,n (¹ m,n 2 ¾ m+n 5) É Â median, É median. 2 Å G H Ô Â, V(H) V(G) ¾ E(H) E(G), H Ô G, Ô H G. Ì H G, ¾ H Â x,y, d H (x,y) = d G (x,y), H É G µ Å. 1 [11] G É Â, u 1,u 2,u 3 V(G), Ì z É u 1,u 2,u 3 Â median, d(u i,z) = 1 2 (d(u i,u j )+d(u i,u k ) d(u j,u k )), ¹ {i,j,k} = {1,2,3}. 2 [11] C É G Ä¾ Ä¾», C G Éµ Å. 3 [15] G É Æ ¾Ú Î». 1 Â median»é Æ. Ñ. G ÉÂ median, G É Æ. C = v 1 v 2...v 2k+1 v 1 É G ÂÄ¾». «2, C G Éµ Å. v 1, v k+1 v k+2, d G (v 1,v k+1 ) = d G (v 1,v k+2 ) = k. z É Â median, «1, d G (v 1,z) = 1 2 (d(v 1,v k+1 )+d(v 1,v k+2 ) d(v k+1,v k+2 )) = 2k 1, 2

3 ÒÊ : median º 3 ¾ ¹ É Â, Å ³ É,. G É Æ. Í«5 G = (V,E) ± G = (V,E ) ÞÚ, É f : V V, Ã uv E, f(u)f(v) E. Í«6 G Ê ϕ É G ± À Â, Ã G u, ϕ 2 (u) = ϕ(u). Ù G ϕ Ç Ü H Ô G ÂÊ. H v, Đ ϕ(v) = v. ÈÃ, v V(H), Ã ϕ(v) = w v. u V(G), Ã ϕ(u) = v. u = w, ϕ 2 (v) = ϕ(ϕ(v)) = ϕ(w) = ϕ(u) = v, ϕ(v) = w, ϕ 2 (v) ϕ(v),. u w, ϕ 2 (u) = ϕ(ϕ(u)) = ϕ(v) = w, ϕ(u) = v, ϕ 2 (u) ϕ(u),. 4 [11] G Ê H É G µ Å. 2 median Ê É median. Ñ G ÉÂ median, H É G Ê. Ü H ÉÂ median. «4, H É G µ Å., H Â u,v w, H median» É G median. z É u,v,w G Â median, P,Q,R ÅÉ G Ä¾ u v, u w, v w. G ± H Ê Ô ϕ, ϕp, ϕq, ϕr H, Ô H Éµ Å, ϕp, ϕq, ϕr ÅÉ H ϕ(u) ϕ(v), ϕ(u) ϕ(w), ϕ(v) ϕ(w) Ä¾. ϕ(z) ÁÖ ϕ(u) ϕ(v), ϕ(u) ϕ(w), ϕ(v) ϕ(w) Ä¾. ϕ(z) É H ϕ(u), ϕ(v), ϕ(w) Â Â median. H, ϕ(u) = u, ϕ(v) = v, ϕ(w) = w, ϕ(z) = z. ¾, z É u,v,w H Â median., H Â» Â median. Ç H ÉÂ median.. Í«7 G 1 = (V 1,E 1 ) G 2 = (V 2,E 2 ), ¹ G 1 G 2 Ð, Ç G 1 G 2, Ü: V(G 1 G 2 ) = V 1 V 2, (u 1,u 2 ) (v 1,v 2 ) Ð ¾Ú u 1 = v 1 ¾ u 2 v 2 E 2, ½ u 1 v 1 E 1 ¾ u 2 = v 2. ÂÀ, ÔßÆ: É V(G 1 G 2 ) ± V 1 p 1 : (u,v) u; É V(G 1 G 2 ) ± V 2 p 2 : (u,v) v. 5 [11] (u,v) (x,y) É G 1 G 2 Â, d G1 G 2 ((u,v),(x,y)) = d G1 (u,x)+d G2 (v,y). Û, Ì Q É (u,v) (x,y) Ä¾, p 1 Q É G 1 u x Ä¾, p 2 Q É G 2 v y Ä¾. 3 median É median. Ñ G 1 G 2 É Â median G 1 G 2, u,v w É G 1 G 2 Â. ¹ z É G 1 G 2 u,v w Â median. Ý z. P,Q R ÅÉ G 1 G 2 Í z u v, u w v w Ä¾. «5, p i z É G i p i u, p i v p i w median, ¹ i {1,2}. Ü Â u,v,w V(G 1 G 2 ), Â¾ median. Ô G 1 G 2»É median, z i É G i p i u, p i v p i w Â median, ¹

4 4 Ü i {1,2}. Þ, d G1 (p 1 u,z 1 )+d G1 (z 1,p 1 v) = d G1 (p 1 u,p 1 v), (1) d G2 (p 2 u,z 2 )+d G2 (z 2,p 2 v) = d G2 (p 2 u,p 2 v). (2) (1)+(2) ³ «3 ³ d G1 (p 1 u,z 1 )+d G1 (z 1,p 1 v)+d G2 (p 2 u,z 2 )+d G2 (z 2,p 2 v) = d G1 (p 1 u,p 1 v)+d G2 (p 2 u,p 2 v). (3) (3) ÅÅ = [d G1 (p 1 u,z 1 )+d G1 (z 1,p 1 v)]+[d G2 (p 2 u,z 2 )+d G2 (z 2,p 2 v)] = [d G1 (p 1 u,z 1 )+d G2 (p 2 u,z 2 )]+[d G1 (z 1,p 1 v)+d G2 (z 2,p 2 v)] ((p 1 u,p 2 u),(z 1,z 2 ))+d G1 G 2 ((z 1,z 2 ),(p 1 v,p 2 v)) (u,z)+d G1 G 2 (z,v), (3) Å = d G1 (p 1 u,p 1 v)+d G2 (p 2 u,p 2 v) ((p 1 u,p 2 u),(p 1 v,p 2 v)) (u,v). É d G1 G 2 (u,z)+d G1 G 2 (z,v) (u,v), z G 1 G 2 Ä¾ u v. «, z G 1 G 2 Ä¾ v w Ä¾ u w., G 1 G 2 u,v w, Â medianz = (z 1,z 2 ). ÇØ. 3 Û G = (V,E) É Â, G 1,G 2 É G Â. G 1,G 2 É G Â (proper cover), V(G 1 ) V(G 2 ) = V(G), V(G 1 ) V(G 2 ) = V(G 0 ), «¾ V(G 1 )\V(G 0 ) V(G 2 )\V(G 0 ) Ð. G È G 1 G 2 É G Æµ Ü G : ; 1) Ò V(G 0 ) v Ð v 1 v 2 ; 2) v 1 v V(G 1 )\V(G 0 )», v 2 v V(G 2 )\V(G 0 )» 3) Ì u,v V(G 0 ) ¾ uv E(G), u 1 v 1, u 2 v 2. G 1 G 2»É G, G É G (convex expansion). ß median quasi-median µ Á,» µ Å [1,16 17]. Mulder [1,18] G É median ¾Ú G Â ÞÍ Û»³±. Ü ØÒÉÙ Û Æ. Ü median É ¾» ØÆ? Ë [1] Mulder, H.M., The Interval Function of a Graph, Amsterdam: Mathematisch Centrum, 1980.

5 ÒÊ : median º 5 [2] Brešar, B., Klavžar, S. and Škrekovski, R., On cube-free median graphs, Discrete Mathematics, 2007, 307(3/4/5): [3] Klavžar, S., Mulder, H.M. and Škrekovski, R., An Euler-type formula for median graphs, Discrete Mathematics, 1998, 187(1): [4] Brešar, B. and Klavžar, S., Maximal proper subgraphs of median graphs, Discrete Mathematics, 2007, 307(11/12): [5] Mulder, H.M., n-cubes and median graphs, Journal of Graph Theory, 1980, 4(1): [6] Mulder, H.M. and Schrijver, A., Median graphs and Helly hypergraphs, Discrete Mathematics, 1979, 25(1): [7] Birkhoff, G. and Kiss, S.A., A ternary operation in distributive lattices, Bulletin of the American Mathematical Society, 1947, 53(8): [8] Avann, S.P., Metric ternary distributive semi-lattices, Proceedings of the American Mathematical Society, 1961, 12(3): [9] Nebeský, L., Median graphs, Commentationes Mathematicae Universitatis Carolinae, 1971, 12(2): [10] Klavžar, S. and Mulder, H.M., Median graphs: characterizations, location theory and related structures, Journal of Combinatorial Mathematics and Combinatorial Computing, 1999, 30: [11] Imrich, W. and Klavžar, S., Product Graphs: Structure and Recognition, New York: John Wiley & Sons, [12] Chung, F.R.K., Graham, R.L. and Saks, M.E., Dynamic search in graphs, In: Discrete Algorithms and Complexity(Wilf, H.S., Nozaki, A., Johnson, D.S. and Takao, N. eds.), Perspectives in Computing, Vol. 15, New York: Academic Press, [13] Imrich, W. and Klavžar S., A convexity lemma and expansion procedures for bipartite graphs, European Journal of Combinatorics, 1998, 19(6): [14] Bandelt, H.J. and Chepoi, V.D., The algebra of metric betweenness II: Geometry and equational characterization of weakly median graphs, European Journal of Combinatorics, 2008, 29(3): [15] West, D.B., Introduction to Graph Theory(Second Edition), New Jersey: Prentice Hall, [16] Bandelt, H.J., Mulder, H.M. and Wilkeit, E., Quasi-median graphs and algebras, Journal of Graph Theory, 1994, 18(7): [17] Chepoi, V.D., Isometric subgraphs of Hamming graphs and d-convexity, Cybernetics and Systems Analysis, 1988, 24(1): [18] Mulder, H.M., The structure of median graphs, Discrete Mathematics, 1978, 24(2): Multimedian Graph and Its Properties WANG Guangfu (School of Basic Science, East China Jiaotong University, Nanchang, Jiangxi, , P. R. China) Abstract: Suppose that u,v and w are three vertices of a graph G. A vertex z is called a median of u,v and w, if the vertex z simultaneously lies on shortest paths between any two of u,v and w. A graph G is called a multimedian graph, if every three vertices of G have at least one median. Some elementary properties are studied. The results show that any multimedian graph is bipartite, the retraction of a multimedian graph is also a multimedian graph, and the product graph of two multimedian graphs is a multimedian graph, too. Key words: shortest path; median; multimedian graph; product graph

Partial Hamming graphs and expansion procedures

Discrete Mathematics 237 (2001) 13 27 www.elsevier.com/locate/disc Partial Hamming graphs and expansion procedures Bostjan Bresar Faculty of Agriculture, University of Maribor, Vrbanska 30, 2000 Maribor,