ON A PROBLEM OF C. E. SHANNON IN GRAPH THEORY

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1 ON A PROBLEM OF C. E. SHANNON IN GRAPH THEORY m. rosefeld1 1. Itroductio. We cosider i this paper oly fiite odirected graphs without multiple edges ad we assume that o each vertex of the graph there is a loop, i.e. each vertex of the graph is coected to itself by a edge. O. Ore i [2] raised the followig problem: Give a fiite graph P, what are the ecessary ad sufficiet coditios o G i order that (1) p(gx H) = p(g) -p(h) for every fiite graph H. A partial aswer is give by the followig theorem due to Shao [3]: Theorem 1 (Shao). // there exists a preservig fuctio a defied o G such that o-(g) is a idepedet set of vertices i G the (1) holds for every fiite graph H. For a proof of Shao's theorem see for example [l], [3]. Shao proved the sufficiecy of his coditio oly. Our mai result is a ecessary ad sufficiet coditio uder which (1) always holds (Theorem 2) ad to show that Shao's coditio is ot ecessary ( 4). Our coditio will be give i terms of liear programmig. 2. Defiitios ad otatios. By a idepedet set of vertices i a graph G we mea a subset of vertices such that o two differet vertices i the subset are joied by a edge i G. The maximal umber of idepedet vertices i a graph G will be deoted by fi(g). A clique i a graph G is a complete subgraph of G (i.e. a set of vertices each pair of which are coected by a edge) which is ot cotaied i ay other complete subgraph of G. Ver(G) will deote the set of vertices of G. A fuctio cr: G >(? will be called preservig if g-t->g' =^o-(g)^jfff(g') (where g-t-»g' meas that g is ot joied by a edge to the vertex g'). The cartesia product of two graphs is a graph deoted by GXff defied as follows: Ver(G X H) = Ver(G) X Ver(P), (gh) -> (g'h') iff g - g' ad h - h'. A graph G for which the equality (1) always holds will be called uiversal. Received by the editors February 3, This paper is a part of the author's Ph.D. thesis to be submitted to the Hebrew Uiversity, Jerusalem. The author wishes to express his gratitude to M. O. Rabi for his helpful suggestios. 315

2 316 M. ROSENFELD [April 3. Let G be a fiite graph. Ver(G) = {gi g}- Let {Ci C,) be a fixed orderig of all the differet cliques of G. Defie af as follows: e =1, Let PG={(X! -x ) Er.i«i0)^^L Theorem gie Cj, = o, giecj. Xi^Q, lrg/rgs}. 2. A fiite graph G is uiversal if ad oly if (2) max 2_, xt = m(g), x = (xi x). z Pq,-_i Proof, (i) Without loss of geerality we may assume that {gi ' ' ' & <«} =A is a idepedet set of vertices i G. Choose: Xi=l, 1 Sitkp.(G), x, = 0, i>(g). Sice o two vertices i A are cotaied i the same clique it is obvious that for every/ we have: "... S ai x* = 1 t=i while 22 xi = p(g). i=i Therefore we always have max zji=i Xi^p(G). (ii) Suppose G is ot uiversal, i.e. there exists a graph H such that p(gx H) > p(g)-p(h). (It is obvious that p(gxh)>(g) -p.(h).) Let ^CGXI^be a maximal idepedet set of vertices i GXH (i.e. card A=p(GXH)). Defie At= {h\ (gih)ea}, (AtCH). Sice (&*)->(&*') if h >h' ad 4 is idepedet it follows that A{ is a idepedet set of vertices i H ad therefore card Ai^p(H). Furthermore if A't { (&k) heat\ the card A=\j"=l A't ad the uio is disjoit. Now choose x( (\/p(h)) card At, it is obvious that A 1 A p(gx H) (3) L *< = "7^ E card 4/ = > p(g).,=i p(i(),=i p.(flj Let us show that, for every/, y,?_, a^'xi^l. If (4) Cj = {gh- gi } =» a,- x,- = X) *.>.=i i=i Sice g.-r»gi,i 1 =?"> 2^&, it follows that Uf=1 A([ is a idepedet set of vertices i H, ad the uio is disjoit, hece the followig holds (H) 2*i= k k / k \ S card ^«i = card<! U ^<if = **(#) (=1 i=l 11=1 '

3 1967] ON A PROBLEM OF C. E. SHANNON IN GRAPH THEORY 317 Thus, (3) ad (4) prove that our coditio is ecessary, (iii) To prove the sufficiecy of our coditio, suppose that max 22 X; > p(g). x&pg,= 1 Sice Pa is a polytope i the ra-dimesioal Euclidea space the maximum is obtaied; furthermore, sice the coefficiets of the half spaces determiig the polytope are oegative itegers we may assume without loss of geerality that all the compoets (xi x ) of the maximizig poit which ca be chose to be a vertex of P<? are ratioal. Let /? be the least commo multiplier of all the deomiators of the x,'s (it is obvious that 0<\ beig the determiat of a matrix of order with O's ad l's). Let y, = /3-Xi, hece {y,} is a set of oegative itegers satisfyig (5) iyi>p(g)-r3. i=i (6) afyi g /3, lgj^. «=i Usig (5) ad (6) we shall costruct a graph H for which the iequality p.(gx H) > p(h)-p(g) will hold. This of course will complete the proof of our theorem. Let Ait 1 ^i^, be a family of disjoit sets such that card A{ = yt. Let Ver H= (J"=1At. Two vertices i H\z, u will be joied by a edge if: (a) z = u (b) z G Ah ue Aj-*i ^ j ad g,- ^ gj. (Hece ay set Ai is idepedet.) Let U= \ui ut\ be a idepedet set of vertices i H. We may assume that Ur\A{^0, l^i^t, ad UC\Ai = 0, i>t. Sice U is idepedet so is U'_i^4i. It follows from our defiitio of H that the set (gi gt\ is a complete subgraph of G ad therefore it is cotaied i a clique of G. Hece we have: 2,,x,^l, But this implies: 22}_iyi^j8=> card {Uj=i^4,} ^/3. This meas that: (7) p(h)^. LetD={(gih)\hEAi} (DCGXH).U (g, h), (g'h')ed, the g-^g' =>&->->&' ad therefore (gh)-^(g'h'); if g-»g' it is obvious that (gh) -*+(g'h') i.e. D is a idepedet set of vertices i GXH. Now usig (5), (6) ad (7) we obtai:

4 318 M. ROSENFELD [April p(g X H) ^ card D = card < U Aii = Y/yi> p(g) -/3 ^ p(g) -p(h). (.,=i ',=i This completes the proof of our theorem. Remarks. The coditio of Theorem 2 ca be expressed i terms of discrete liear programmig as follows: G is uiversal if ad oly if for ay set of oegative itegers x, satisfyig: E «*0)«* ^ ft i ^ i ^ *, E *< ^ m(g) -/3 for all oegative itegers /?. Suppose G is ot uiversal, i.e. there exists for which maxxxi Xi>fi(G) /?. If {gi g cg)lis a idepedet set of vertices i G, choose yi = x,-f-l (l^i^fi(g)), y,=x,- (i>pi(g)); it is obvious that 2_?-i<*Wy<^/H-l while?_,?<>/i(6)-(j8+l). This shows that if G is ot uiversal with respect to /3 it is also ot uiversal with respect to j8+l. Sice the umber of differet graphs (up to isomorphism) with vertices is fiite, it follows that there exists a iteger p() such that G is uiversal if ad oly if G is uiversal with respect to /?(«). The fuctio fi() is a odecreasig fuctio of. The values of /3(«) for «^5 may be easily computed usig Shao's observatio [3] that all graphs with at most 5 vertices are uiversal except for the petago which is ot uiversal with respect to 2. Hece fi() =0, 5S4, 0(5) = 2. Usig (Hi) i the proof of Theorem 2 oe ca see that fi() <\. Oe ca use Theorem 2 to estimate the value of p.(gxh) as follows: give G ad H oe ca calculate subject to a = max ^ *< E'*< = m(s), l^j^ic, <=1 where x,- is a oegative iteger ad b = max E*i y t-i ft0)y.- ^m(g), t j g sff. (ftw has the same meaig with respect to If as a^ with respect It is obvious that: ju(gx.ff)^mi (a, b). to G.) 4. I this paragraph we shall show that Shao's coditio is ot ecessary. Observe first that if G is a graph ad a a preservig fuctio defied o G, ad if A CG is a idepedet set of vertices, the

5 1967] ON A PROBLEM OF C. E. SHANNON IN GRAPH THEORY 319 a(a) is idepedet ad cara\{o-(a)\ =card A; therefore we always have (<r(g)) =(G). Sice <r~l(g) is a complete subgraph of G it follows that G is covered by card{ver a(g)\ complete subgraphs, therefore a ecessary coditio for the existece of a preservig fuctio a such that a(g) is idepedet is that G is covered by p.(g) complete subgraphs. Let Gi ad G2 be two disjoit petagos ad G3 a set of 5 vertices o oe of which belogs to Gi or G2. Adjoi by a edge each vertex of G3 to all the vertices of Gi ad G2. Let H be the graph defied by these relatios, hece we have: cardjverp} = 15, p(h) = 5 (G3 is idepedet). Sice a petago caot be covered by less tha 3 complete subgraphs, it is obvious that H caot be covered by less tha 6 complete subgraphs. Thus we have show that Shao's coditio caot hold for H. To show that H is uiversal observe that all the cliques of H are triagles, every vertex of H is cotaied i exactly 10 differet cliques ad the umber of differet cliques is 50, therefore the followig holds: but: ^ j ^ 50, ctvxi ^ 1 ^ Z E «.1 Xi g 50, t-i y-i t-i X) H «< x4 = 10 Xi ^ 50 =* max ^ *» = 5 = /*(#) j=i 1=1»=i 1=1 ad, by Theorem 2, H is uiversal. Refereces Q.E.D. 1. C. Berge, Theorie des graphes et ses applicatios, Duod, Paris, O. Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ., Vol. 38, Amer. Math. Soc, Providece, R. I., C. E. Shao, The zero error capacity of a oisy chael, I.R.E., Trasactios o Iformatio Theory, IT-2, Hebrew Uiversity