THE COMPETITION NUMBERS OF JOHNSON GRAPHS

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1 Discussioes Mathematicae Graph Theory 30 (2010 ) THE COMPETITION NUMBERS OF JOHNSON GRAPHS Suh-Ryug Kim, Boram Park Departmet of Mathematics Educatio Seoul Natioal Uiversity, Seoul , Korea Yoshio Sao Research Istitute for Mathematical Scieces Kyoto Uiversity, Kyoto , Japa Abstract The competitio graph of a digraph D is a graph which has the same vertex set as D ad has a edge betwee two distict vertices x ad y if ad oly if there exists a vertex v i D such that (x, v) ad (y, v) are arcs of D. For ay graph G, G together with sufficietly may isolated vertices is the competitio graph of some acyclic digraph. The competitio umber k(g) of a graph G is defied to be the smallest umber of such isolated vertices. I geeral, it is hard to compute the competitio umber k(g) for a graph G ad to characterize all graphs with give competitio umber k has bee oe of the importat research problems i the study of competitio graphs. The Johso graph J(, d) has the vertex set {v X X ( ) [] }, ) d deotes the set of all d-subsets of a -set [] = {1,..., }, where ( [] d ad two vertices v X1 ad v X2 are adjacet if ad oly if X 1 X 2 = d 1. I this paper, we study the edge clique umber ad the competitio umber of J(, d). Especially we give the exact competitio umbers of J(, 2) ad J(, 3). This work was supported by the Korea Research Foudatio Grat fuded by the Korea Govermet (MOEHRD) (KRF C00004). The author was supported by Seoul Fellowship. Correspodig author. address: kawa22@su.ac.kr The author was supported by JSPS Research Fellowships for Youg Scietists. The author was also supported partly by Global COE program Fosterig Top Leaders i Mathematics.

2 450 S.-R. Kim, B. Park ad Y. Sao Keywords: competitio graph, competitio umber, edge clique cover, Johso graph Mathematics Subject Classificatio: 05C69, 05C Itroductio The competitio graph C(D) of a digraph D is a simple udirected graph which has the same vertex set as D ad has a edge betwee two distict vertices x ad y if ad oly if there is a vertex v i D such that (x, v) ad (y, v) are arcs of D. The otio of a competitio graph was itroduced by Cohe [3] as a meas of determiig the smallest dimesio of ecological phase space (see also [4]). Sice the, various variatios have bee defied ad studied by may authors (see [11, 15] for surveys ad [1, 2, 7, 8, 9, 10, 12, 14, 19, 20] for some recet results). Besides a applicatio to ecology, the cocept of competitio graph ca be applied to a variety of fields, as summarized i [17]. Roberts [18] observed that, for a graph G, G together with sufficietly may isolated vertices is the competitio graph of a acyclic digraph. The he defied the competitio umber k(g) of a graph G to be the smallest umber k such that G together with k isolated vertices is the competitio graph of a acyclic digraph. A subset S of the vertex set of a graph G is called a clique of G if the subgraph of G iduced by S is a complete graph. For a clique S of a graph G ad a edge e of G, we say e is covered by S if both of the edpoits of e are cotaied i S. A edge clique cover of a graph G is a family of cliques such that each edge of G is covered by some clique i the family. The edge clique cover umber θ E (G) of a graph G is the miimum size of a edge clique cover of G. We call a edge clique cover of G with the miimum size θ E (G) a miimum edge clique cover of G. A vertex clique cover of a graph G is a family of cliques such that each vertex of G is cotaied i some clique i the family. The vertex clique cover umber θ V (G) of a graph G is the miimum size of a vertex clique cover of G. Dutto ad Brigham [5] characterized the competitio graphs of acyclic digraphs usig edge clique covers of graphs. Roberts [18] observed that the characterizatio of competitio graphs is equivalet to the computatio of competitio umbers. It does ot seem to be easy i geeral to compute k(g) for a graph G, as Opsut [16] showed

3 The Competitio Numbers of Johso Graphs 451 that the computatio of the competitio umber of a graph is a NP-hard problem (see [11, 13] for graphs whose competitio umbers are kow). For some special graph families, we have explicit formulae for computig competitio umbers. For example, if G is a chordal graph without isolated vertices the k(g) = 1, ad if G is a otrivial triagle-free coected graph the k(g) = E(G) V (G) + 2 (see [18]). I this paper, we study the competitio umbers of Johso graphs. We deote a -set {1,..., } by [] ad the set of all d-subsets of a -set by ( []) d. The Johso graph J(, d) has the vertex set {vx X ( []) d }, ad two vertices v X1 ad v X2 are adjacet if ad oly if X 1 X 2 = d 1 (for referece, see [6]). For example, the Johso graph J(5, 2) is give i Figure 1. v {1,3} v {1,2} PSfrag replacemets v {1,5} v {2,4} v {2,5} v {2,3} v {4,5} v {3,5} v{1,4} v {3,4} Figure 1. The Johso graph J(5, 2). As it is kow that J(, d) = J(, d), we assume that 2d. Our mai results are the followig. Theorem 1. For 4, we have k(j(, 2)) = 2. Theorem 2. For 6, we have k(j(, 3)) = 4. We use the followig otatio ad termiology i this paper. For a digraph D, a orderig v 1, v 2,..., v of the vertices of D is called a acyclic orderig of D if (v i, v j ) A(D) implies i < j. It is well-kow that a digraph D is acyclic if ad oly if there exists a acyclic orderig of D. For a digraph D ad a vertex v of D, the out-eighborhood of v i D is the set {w V (D) (v, w) A(D)}. A vertex i the out-eighborhood of a vertex v i a digraph D is called a prey of v i D. For simplicity, we deote the

4 452 S.-R. Kim, B. Park ad Y. Sao out-eighborhood of a vertex v i a digraph D by P D (v) istead of usual otatio N D + (v). For a graph G ad a vertex v of G, we defie the (ope) eighborhood N G (v) of v i G to be the set {u V (G) uv E(G)}. We sometimes also use N G (v) to stad for the subgraph iduced by its vertices. 2. A Lower Boud for the Competitio Number of J(, d) I this sectio, we give lower bouds for the competitio umber of the Johso graph J(, d). Lemma 3. Let ad d be positive itegers with 2d. For ay vertex x of the Johso graph J(, d), we have θ V (N J(,d) (x)) = d. P roof. If d = 1, the J(, d) is a complete graph ad the lemma is trivially true. Assume that d 2. Take ay vertex x i J(, d). The x = v A for some A ( []) d. For ay vertex va i J(, d), the set S i (v A ) := {v B B = (A \ {i}) {j} for some j [] \ A} forms a clique of J(, d) for each i A. To see why, take two distict vertices v B ad v C i S i (v A ). The B = (A \ {i}) {j} ad C = (A \ {i}) {k} for some distict j, k [] \ A. Clearly B C = d 1, ad so v B ad v C are adjacet i J(, d). Take a vertex v B i N J(,d) (v A ). The B = (A \ {i}) {j} for some i A ad j [] \ A ad so v B S i (v A ). Thus {S i (v A ) i A} is a vertex clique cover of N J(,d) (v A ). Thus θ V (N J(,d) (v A )) d. O the other had, ((A \ {i}) {j}) ((A \ {i }) {j }) = d 2 if i, i A ad j, j [] \ A satisfy i i ad j j (such i, i, j, j exist sice 2d 4). This implies that θ V (N J(,d) (v A )) d. Hece θ V (N J(,d) (v A )) = d. Opsut [16] gave a lower boud for the competitio umber of a graph G as follows: k(g) mi{θ V (N G (v)) v V (G)}. Together with Lemma 3, we have k(j(, d)) d for positive itegers ad d satisfyig 2d. The followig theorem gives a better lower boud for k(j(, d)) if d 2.

5 The Competitio Numbers of Johso Graphs 453 Theorem 4. For 2d 4, we have k(j(, d)) 2d 2. P roof. Put k := k(j(, d)). The there exists a acyclic digraph D such that C(D) = J(, d) I k, where I k = {z 1, z 2,..., z k } is a set of isolated vertices. Let x 1, x 2,..., x ( d), z 1, z 2,..., z k be a acyclic orderig of D. Let v 1 := x ( d) ad v 2 := x ( d) 1. By Lemma 3, we have θ V (N J(,d) (x i )) = d for i = 1,..., ( d). Thus vi has at least d distict prey i D, that is, (2.1) P D (v i ) d. Sice x 1, x 2,..., x ( d), z 1, z 2,..., z k is a acyclic orderig of D, we have (2.2) P D (v 1 ) P D (v 2 ) I k {v 1 }. Moreover, we may claim the followig: Claim. For ay two adjacet vertices v X1 ad v X2 of J(, d), we have P D (v X1 ) \ P D (v X2 ) d 1. Proof of Claim. Suppose that v X1 ad v X2 are adjacet i J(, d). The X 1 X 2 = d 1 ad [] \ (X 1 X 2 ) 2d X 1 X 2 + X 1 X 2 = d 1. We take d 1 elemets from [] \ (X 1 X 2 ), say z 1, z 2,..., z d 1, ad put X 1 X 2 := {y 1, y 2,..., y d 1 }. For each 1 j d 1, we put Z j := X 1 {z j }\{y j }. The Z j = d ad so v Zj is a vertex i J(, d). Note that Z j X 1 = d 1 ad Z j X 2 = d 2. Thus v Zj is adjacet to v X1 while it is ot adjacet to v X2. Therefore This implies P D (v X1 ) P D (v Zj ) ad P D (v X2 ) P D (v Zj ) =. (2.3) P D (v X1 ) \ P D (v X2 ) d 1 j=1 ( PD (v X1 ) P D (v Zj ) ), ad, trivially, for each j {1,..., d 1}, (2.4) P D (v X1 ) P D (v Zj ) 1.

6 454 S.-R. Kim, B. Park ad Y. Sao Note that Z j Z i = d 2 for i j. Therefore v Zi ad v Zj ad so P D (v Zi ) P D (v Zj ) =. Thus, for i j, are ot adjacet (2.5) (P D (v X1 ) P D (v Zi )) ( P D (v X1 ) P D (v Zj ) ) =. From (2.3), (2.4), ad (2.5), it follows that d 1 P D (v X1 ) \ P D (v X2 ) P D (v X1 ) P D (v Zj ) d 1. j=1 This completes the proof of the claim. Now suppose that v 1 ad v 2 are ot adjacet i J(, d). The v 1 ad v 2 do ot have a commo prey i D, that is, (2.6) P D (v 1 ) P D (v 2 ) =. By (2.1), (2.2) ad (2.6), we have k + 1 P D (v 1 ) P D (v 2 ) = P D (v 1 ) + P D (v 2 ) 2d. Hece k 2d 1 > 2d 2. Next suppose that v 1 ad v 2 are adjacet i J(, d). The v 1 ad v 2 have at least oe commo prey i D, that is, (2.7) P D (v 1 ) P D (v 2 ) 1. By the above claim, (2.8) P D (v 1 ) \ P D (v 2 ) d 1 ad P D (v 2 ) \ P D (v 1 ) d 1. The k + 1 P D (v 1 ) P D (v 2 ) (by (2.2)) = P D (v 1 ) \ P D (v 2 ) + P D (v 2 ) \ P D (v 1 ) + P D (v 1 ) P D (v 2 ) (d 1) + (d 1) + 1 (by (2.7) ad (2.8)) = 2d 1. Hece it holds that k 2d 2.

7 The Competitio Numbers of Johso Graphs A Miimum Edge Clique Cover of J(, d) I this sectio, we build a miimum edge clique cover of J(, d). Give a Johso graph J(, d), we defie a family Fd of cliques of J(, d) as follows. For each Y ( [], we put Note that S Y S Y := {v X X = Y {j} for j [] Y }. is a clique of J(, d) with size d + 1. We let (3.1) F d := {S Y Y ( ) [] }. d 1 The it is ot difficult to show that Fd is the collectio of cliques of maximum size. Moreover the family Fd is a edge clique cover of J(, d). To see why, take ay edge v X1 v X2 of J(, d). The X 1 X 2 = d 1 ad both of v X1 ad v X2 belog to the clique S X1 X 2 Fd. Thus F d is a edge clique cover of J(, d). We will show that Fd is a miimum edge clique cover of J(, d). Prior to that, we preset the followig theorem. For two distict cliques S ad S of a graph G, we say S ad S are edge disjoit if S S 1. Theorem 5. θ E (J(, d)) = ( ad ay miimum edge clique cover of J(, d) cosists of edge disjoit maximum cliques. P roof. Let E be a miimum edge clique cover for J(, d), that is, θ E (J(, d)) = E. Sice Fd is a edge clique cover with F d = (, we have θ E (J(, d)) (. Now we show that E (. Sice the size of a maximum clique is d + 1, we have E(S) ( d+1) ( 2 for each S E where E(S) = S ) 2. Therefore, (3.2) E(J(, d)) ( ) d + 1 E(S) E, 2 S E ad the first equality holds if ad oly if oe of two distict cliques i E have a commo edge, ad the secod equality holds if ad oly if ay elemet of E is a maximum clique i J(, d).

8 456 S.-R. Kim, B. Park ad Y. Sao Sice the Johso graph J(, d) is a d( d)-regular graph ad the umber of vertices of J(, d) is ( d), (3.3) E(J(, d)) = 1 ( ) ( ) ( ) d d( d) =. d 2 d 1 From (3.2) ad (3.3), it follows that ( ) ( d+1 2 ) ( d 1 d+1 ) 2 E. Thus we have ( ) d 1 E. Hece we ca coclude that θe (J(, d)) = (. Furthermore, two equalities i (3.2) must hold, ad therefore ay miimum edge clique cover of J(, d) cosists of edge disjoit maximum cliques. Sice F d = (, the followig corollary is a immediate cosequece of Theorem 5: of J(, d) defied i (3.1) is a mi- Corollary 6. The edge clique cover Fd imum edge clique cover of J(, d). 4. Proofs of Theorems 1 ad 2 First, we defie a order o the set ( []) d as follows. Take two distict elemets X 1 ad X 2 i ( []) d. Let X1 = {i 1, i 2,..., i d } ad X 2 = {j 1, j 2,..., j d } where i 1 < < i d ad j 1 < < j d. The we defie X 1 X 2 if there exists t {1,..., d} such that i s = j s for 1 s t 1 ad i t < j t. It is easy to see that is a total order. Now we prove Theorem 1. Proof of Theorem 1. As k(j(, 2)) 2 by Theorem 4, it remais to show k(j(, 2)) 2. We defie a digraph D as follows: where I 2 = {z 1, z 2 }, ad A(D) = 2 i=1 V (D) = V (J(, 2)) I 2 { (x, v{i+1,i+2} ) x S {i} F 2 } 2 { (x, zi ) x S { 2+i} F2 }. i=1

9 The Competitio Numbers of Johso Graphs 457 Sice the vertices of each clique i the edge clique cover F2 has a commo prey i D, it holds that C(D) = J(, 2) I 2. Each vertex i S {i} is deoted by v X for some X ( []) 2 which cotais i. The by the defiitio of, v X v {i+1,i+2} for i = 1,..., 2. Thus, there exists a arc from a vertex x to a vertex y i D if ad oly if either x = v X ad y = v Y with X Y, or x = v X ad y = z i with X S { 1} S {} ad i {1, 2}. Therefore D is acyclic. Thus we have k(j(, 2)) 2 ad this completes the proof. Proof of Theorem 2. By Theorem 4, we have k(j(, 3)) 4. It remais to show k(j(, 3)) 4. We defie a digraph D as follows: where I 4 = {z 1, z 2, z 3, z 4 }, ad V (D) = V (J(, 3)) I 4 A(D) = 3 2 i=1 j=i+1 3 i=1 4 { (x, v{i,j+1,j+2} ) x S {i,j} F 3 } { (x, v{i+1,i+2,i+3} ) x S {i, 1} F 3 } { (x, v{i+1,i+2,i+4} ) x S {i,} F3 } i=1 3 {(x, z i ) x S { 4+i,} F3 } i=1 {(x, z 4 ) x S { 2, 1} F 3 }. It is easy to check that F3 = {S {i,j} i = 1,..., 3; j = i + 1,..., 2} {S {i, 1} i = 1,..., 3} {S {i,} i = 1,..., 4} {S { 3,}, S { 2,}, S { 1,} } {S { 2, 1} }. Thus C(D) = J(, 3) I 4. Moreover, ay vertex x S {i,j} is deoted by v X for some X ( []) 3 which cotais i ad j. By the defiitio of,

10 458 S.-R. Kim, B. Park ad Y. Sao X {i, j+1, j+2}. I a similar maer, for x i other cliques i F3, we may show that (x, y) A(D) if ad oly if either x = v X ad y = v Y with X Y, or x = v X ad y = z i with X S { 3,} S { 2,} S { 1,} S { 2, 1} ad i {1, 2, 3, 4}. Thus D is acyclic. Hece k(j(, 3)) Cocludig Remarks I this paper, we gave some lower bouds for the competitio umbers of Johso graphs, ad computed the competitio umbers of Johso graphs J(, 2) ad J(, 3). It would be atural to ask: What is the exact value of the competitio umber of a Johso graph J(, 4) for 8? Evetually, what are the exact values of the competitio umbers of the Johso graphs J(, q) for q 5? Refereces [1] H.H. Cho ad S.-R. Kim, The competitio umber of a graph havig exactly oe hole, Discrete Math. 303 (2005) [2] H.H. Cho, S.-R. Kim ad Y. Nam, O the trees whose 2-step competitio umbers are two, Ars Combi. 77 (2005) [3] J.E. Cohe, Iterval graphs ad food webs: a fidig ad a problem, Documet PR, RAND Corporatio (Sata Moica, CA, 1968). [4] J.E. Cohe, Food webs ad Niche space (Priceto Uiversity Press, Priceto, NJ, 1978). [5] R.D. Dutto ad R.C. Brigham, A characterizatio of competitio graphs, Discrete Appl. Math. 6 (1983) [6] C. Godsil ad G. Royle, Algebraic Graph Theory, Graduate Texts i Mathematics 207 (Spriger-Verlag, 2001). [7] S.G. Hartke, The elimiatio procedure for the phylogey umber, Ars Combi. 75 (2005) [8] S.G. Hartke, The elimiatio procedure for the competitio umber is ot optimal, Discrete Appl. Math. 154 (2006) [9] G.T. Helleloid, Coected triagle-free m-step competitio graphs, Discrete Appl. Math. 145 (2005) [10] W. Ho, The m-step, same-step, ad ay-step competitio graphs, Discrete Appl. Math. 152 (2005)

11 The Competitio Numbers of Johso Graphs 459 [11] S.-R. Kim, The competitio umber ad its variats, i: Quo Vadis, Graph Theory, (J. Gimbel, J.W. Keedy, ad L.V. Quitas, eds.), Aals of Discrete Mathematics 55 (North-Hollad, Amsterdam, 1993) [12] S.-R. Kim, Graphs with oe hole ad competitio umber oe, J. Korea Math. Soc. 42 (2005) [13] S.-R. Kim ad F.S. Roberts, Competitio umbers of graphs with a small umber of triagles, Discrete Appl. Math. 78 (1997) [14] S.-R. Kim ad Y. Sao, The competitio umbers of complete tripartite graphs, Discrete Appl. Math. 156 (2008) [15] J.R. Ludgre, Food Webs, Competitio Graphs, Competitio-Commo Eemy Graphs, ad Niche Graphs, i: Applicatios of Combiatorics ad Graph Theory to the Biological ad Social Scieces, IMH Volumes i Mathematics ad Its Applicatio 17 (Spriger-Verlag, New York, 1989) [16] R.J. Opsut, O the computatio of the competitio umber of a graph, SIAM J. Algebraic Discrete Methods 3 (1982) [17] A. Raychaudhuri ad F.S. Roberts, Geeralized competitio graphs ad their applicatios, Methods of Operatios Research, 49 (Ato Hai, Köigstei, West Germay, 1985) [18] F.S. Roberts, Food webs, competitio graphs, ad the boxicity of ecological phase space, i: Theory ad applicatios of graphs (Proc. Iterat. Cof., Wester Mich. Uiv., Kalamazoo, Mich., 1976) (1978) [19] F.S. Roberts ad L. Sheg, Phylogey umbers for graphs with two triagles, Discrete Appl. Math. 103 (2000) [20] M. Sotag ad H.-M. Teichert, Competitio hypergraphs, Discrete Appl. Math. 143 (2004) Received 14 April 2009 Revised 9 October 2009 Accepted 10 October 2009