Note on p-competition Graphs and Paths

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1 Southeast Asian Bulletin of Mathematics (2018) 42: Southeast Asian Bulletin of Mathematics c SEAMS Note on p-competition Graphs and Paths Y. Kidokoro, K. Ogawa, S. Tagusari, M. Tsuchiya Department of Mathematical Sciences, Tokai University Hiratsuka , Japan mtsuchiya@tsc.u-tokai.ac.jp Received 27 July 2016 Accepted 14 March 2017 Communicated by K. Denecke AMS Mathematics Subject Classification(2000): 05C20, 05C75 Abstract. The p-competition graph C p(d) of a digraph D = (V,A) is a graph with V(C p(d)) = V(D), where an edge between distinct vertices x and y if and only if there exist p distinct vertices,,...,v p V such that x v i, y v i are arcs of the digraph D for each i = 1,2,...,p. In this paper, we obtain that a path with order n 6 is a p-competition graph if and only if n p + 3. We also show that for a p-competition graph G with no isolated vertices, G+K n is a p+n-competition graph. And we obtain that a wheel W n is a p-competition graph if and only if n p+3. Keywords: p-competition graph; Path; Wheel. 1. Introduction In this paper we consider finite simple graphs without loops and finite digraphs allowed loops. For a graph G and S V(G), S V is the induced subgraph of S. A clique in a graph G is the vertex set of a maximal complete subgraph of G. A family Q = {Q 1,Q 2,...,Q l } is an edge clique cover of G if each Q i is a clique of G and for each uv E(G), there exists Q i Q such that u,v Q i. For a digraph D and v V(D), Out D (v) = {u;v u A(D)} and In D (v) = {u;u v A(D)}. Definition 1.1. For a digraph D, the p-competition graph C p (D) of D is the Corresponding author.

2 576 Y. Kidokoro et al. graph satisfying the following: (i) V(C p (D)) = V(D), (ii) xy E(C p (D)) if and only if there exist distinct p vertices,,..., v p V(D) such that x v i, y v i A(D) for each i = 1,2,...,p. AgraphGiscalledap-competition graphifthereexistsadigraphd suchthat C p (D) = G. Let ( [r] p) be the family of all p-subsets of an r-set [r] = {1,2,...,r}. v 0 v 5 v 4 Digraph D v 0 v 5 v 4 2-competition graph C 2 (D) Figure 1: A digraph D and the 2-competition graph C 2 (D) Definition 1.2. (p-edge Clique Cover) Let G be a graph and F = {S 1, S 2,..., S r } be a multifamily of subsets of the vertex set of G. The multifamily F is called a p-edge clique cover if it satisfies the following: (i) For any J ( ) [r] p, the set j J S j is a clique of G or an empty set, (ii) The family { j J S j J ( [r] p) } is an edge clique cover of G. Then θe(g) p = min F:p edge clique cover of G F is called the p-edge clique cover number of G. Dutton and Brigham [3] and Roberts and Steif [13] gave characterizations of competition graphs, which are 1-competition graphs. Lundgren [9] dealt with several properties of competition graphs. In [5] Isaak et al. gave the result on 2-competition graphs, that is, K 2,n is the 2-competition if and only if n = 1 or n 9. Jacobson [6] also gave the result on 2-competition graphs, that is, K n,n is the 2-competition if and only if n = 1. Anderson et al. [1] and Lundgren et al. [10] dealt with p-competition graphs of special types of digraphs. In [11] McKee and McMorris surveyed various properties of intersection graphs, which contain p-competition graphs. Kim et al. [7] gave a characterization of p-completion graphs in terms of p-edge clique covers as follows.

3 Note on p-competition Graphs and Paths 577 v 0 Graph G v 5 v 4 S 1 = {v 0,,v 4 } S 2 = {,,v 5 } S 3 = {, } S 4 = {v 0,,,,v 4,v 5 } S 5 = {v 0,,,,v 4,v 5 } F = {S 1,S 2,S 3,S 4,S 5 } θe 3(G) = 5 Figure 2: A graph G and a 3-edge clique cover F of G Theorem 1.3. [7] For a graph G with order n, G is a p-competition graph if and only if θ p e(g) n. Kim et al. [7] also gave the following result on chordal graphs. They dealt with individual graphs such as chordal graphs. They estimated θ 2 e(g) of individual graphs. Theorem 1.4. [7] (i) A chordal graph is a 2-competition graph, (ii) A cycle C 4 is not a 2-competition graph, (iii) For n > 4, a cycle C n is a 2-competition graph. In [8] Kim et al. dealt with cycles in terms of p-competition graphs. Kim et al. estimated θ p e (C n) and gave a characterization of p-competition cycles as follows. Theorem 1.5. [8] Let C n be a cycle with order n 5 and p be a positive integer. Then C n is a p-competition graph if and only if n p+3. In [2] Chen et al. obtained some properties of paths in bipartite graphs. They dealt with neighborhood conditions for vertex-disjoint paths in bipartite graphs. Using neighborhood conditions of the corresponding digraphs, we study p-competition graphs. Since each path is a chordal graph, we have the following result by Theorem 1.4. Corollary 1.6. Let P n be a path with order n 2. Then P n is a 2-competition graph.

4 578 Y. Kidokoro et al. In this paper we deal with paths in terms of p-completion graphs. In [12] Ogawa et al. considered on line graphs, that is, a unary graph operation. And Era et al. [4] dealt with upper bound graphs and double bound graphs in terms of special vertices, that is, universal vertices. In this paper we consider a sum operation on p-competition graphs. And we show some properties of wheels on p-competition graphs. A wheel is a sum graph of a cycle and an isolated vertex. 2. Paths In this section we consider a path with n vertices in terms of p-completion graphs. Let P n be a path with n vertices: V(P n ) = {v 0,,...,v n 1 } and E(P n ) = {v i v i+1 i = 0,1,...,n 2}. First we obtain the following result. Proposition 2.1. Paths P 2, P 3 and P 4 are not 3-competition graphs. Proof. First we consider the case P 2. Since the vertex number of P 2 is two, P 2 is not a 3-competition graph. Next we consider the case P 3. We assume that P 3 is a 3-competition graph of a digraph D with V(D) = {v 0,, }. Since v 0 E(P 3 ), the digraph D has arcs v 0 v 0, v 0, v 0, v 0, and. Since E(P 3 ), there exist arcs v 0,,, v 0, and. Then we consider six arcs v 0 v 0, v 0, v 0,, v 0, and. Thus v 0 and are adjacent in the 3-competition graph of D, which is a contradiction. We consider the case P 4. We assume that P 4 is a 3-competition graph of a digraph D with V(D) = {v 0,,, }. Since v 0 E(P 4 ), Out(v 0 ) Out( ) has three vertices of v 0,, and. Case 1: Out(v 0 ) Out( ) = {v 0,, }. Since E(P 4 ), we assume that there exist arcs v 0,,, v 0, and. Then for v 0 and, we have six arcs v 0 v 0, v 0, v 0,, v 0 and. Thus v 0 and are adjacent in the 3-competition graph of D, which is a contradiction. Hence Out( ) Out( ) is {v 0,, }, {v 0,, }, or {,, }. Then Out( ) = {v 0,,, }. Since E(P 4 ),Out( ) Out( )is{v 0,, }, {v 0,, }, {v 0,, }, or {,, }. Since Out( ) = {v i i = 0,1,2,3}, Out( ) Out( ) = 3. Thus and are adjacent in the 3-competition graph of D, which is a contradiction. Case 2: Out(v 0 ) Out( ) = {v 0,, }. As is the case with Case 1, we have a contradiction. Case 3: Out(v 0 ) Out( ) = {v 0,, }. Since E(P 4 ), we assume that there exist arcs v 0,,, v 0, and. Then for v 0 and, we have six arcs v 0 v 0, v 0, v 0,, v 0 and. Thus v 0 and

5 Note on p-competition Graphs and Paths 579 are adjacent in the 3-competition graph of D, which is a contradiction. Hence Out( ) Out( ) is {v 0,, }, {v 0,, }, or {,, }. Then Out( ) = {v 0,,, }. So Out( ) Out( ) = 3. Thus and are adjacent in the 3-competition graph of D, which is a contradiction. Case 4: Out(v 0 ) Out( ) = {,, }. As is the case with Case 3, we have a contradiction. Using similar arguments of the proof of Proposition 2.1, we obtain that P 5 is not 3-competition graphs. The following theorem is a generalization of Corollary 1.6 and Proposition 2.1. Theorem 2.2. Let P n be a path with order n 2 and p be a positive integer. Then P n is a p-competition graph if and only if n p+3. Proof. We assume that P n is a p-competition graph, p is a positive integer and p n 2. Let D be a digraph whose p-competition graph is P n, that is, C p (D) = P n, where V(D) = {v 0,,...,v n 1 }. Let F = {In(v 0 ),In( ),...,In(v n 1 )}. Since Out(v i ) Out(v j ) p for v i v j E(G), {In(v k ) F v i,v j In(v k )} p. Thus for any edge v i v j E(G), there exists the set k J In(v k) which contains both v i and v j, where J ( ). For each vertex pair p ), Out(u) Out(w) contains u and w of k J In(v k), where J ( p {v k k J} and Out(u) Out(w) p. Thus k J In(v k) is a clique of G. Therefore { k J In(v k) J ( ) p } is an edge clique cover of Pn and F = {In(v 0 ),In( ),...,In(v n 1 )} is a p-edge clique cover of P n. We assume that for i 0,n 1, v i V(P n ) belongs to all In(v k ) F. Since v i+1 and v i+2 are adjacent in P n, there exists the set k J In(v k) which contains v i+1 and v i+2, where J ( ) p. Then this set k J In(v k) contains v i. So v i is adjacent to v i+2, which is a contradiction. Thus v i belongs to at most n 1 sets of F. Since v i and v i+1 are adjacent in P n and v i 1 and v i are adjacent in P n, there exists the set k J In(v k) which contains v i and v i+1, and the set k K In(v k) which contains v i 1 and v i, where J,K ( ) p. On the other hand, since v i 1 are not adjacent to v i+1, the number of sets of F which contain v i 1 and v i+1 is at most p 1. ThusthenumberofsetsofF whichcontainv i isatleast(p+p) (p 1) = p+1. So we have that p+1 { In(v k ) F v i In(v k ) } n 1. Since p n 2, p + 1 n 1 and { In(v k ) F v i In(v k ) } = n 1. We also obtain { In(v k ) F v i+1 In(v k ) } = n 1 and { In(v k ) F v i+2 In(v k ) } = n 1. Since v i+1 is adjacent to v i, v i+2 and v i is not adjacent to v i+2, {v i,v i+1 } belongs to at most n 2 common sets in F. So p = n 2, because p n 2. Then {v i,v i+2 } belongs to at least n 2 = p common sets in F, because F is n. Thus v i and v i+2 are adjacent, which is a contradiction. Therefore n p+3. Next we show that P n is a p-competition graph for n p+3. We construct

6 580 Y. Kidokoro et al. a digraph D with order n as follows: (i) V(D) = {v 0,,...,v n 1 }, (ii) A(D) = {v 0 v i i = 1,2,...,p} {v n 1 v n 1, v n 1 v i (i = 0,1,...,p 2)} {v i v i, v i v i+1, v i v i+p (i = 1,2,...,n 2) }, where any subscripts are reduced to modulo n. v 0 v 1 v 5 v 2 v 4 Digraph D v 0 v 5 v 4 2-competition graph P 6 Figure 3: For p = 2 and n = 6, a digraph D and the 2-competition graph C 2 (D) = P 6 Then Out(v 0 ) = {,,...,v p }, Out(v n 1 ) = {v n 1,v 0,...,v p 2 }, Out(v i ) = {v i,v i+1,...,v i+p }, where any subscripts are reduced to modulo n. Then for Out(v i ) and Out(v i+1 ) (i = 1,2,...,n 3), Out(v i ) Out(v i+1 ) = {v i+1,v i+2,...,v i+p } and Out(v i ) Out(v i+1 ) = p. Thus the p-competition graph C p (D) has edges {v i,v i+1 } (i = 1,2,...,n 3). Then Out(v 0 ) Out( ) = {,,...,v p } and Out(v 0 ) Out( ) = p. Also Out(v n 2 ) Out(v n 1 ) = {v n 1,v 0,...,v p 2 } and Out(v n 2 ) Out(v n 1 ) = p. Thus the p-competition graph C p (D) has edges {v 0, }, {, },..., {v n 2,v n 1 }. Since Out(v n 1 ) Out(v 0 ) = {,,...,v p 2 } and Out(v n 1 ) Out(v 0 ) = p 2 < p, v n 1 is not adjacent to v 0 in the p-competition graph C p (D). For Out(v 0 ) = {,,...,v p } and Out(v i ) = {v i,v i+1,...,v i+p }, Out(v 0 ) Out(v i ) p 1 for i = 2,3,...,n 2. Thus v 0 is not adjacent to each v i in the p-competition graph C p (D) for i = 2,3,...,n 2. Since n p+3 and Out(v n 1 ) = p, Out(v n 1 ) Out(v i ) p 1 and v n 1 isnot adjacenttoeachv i inthe p-competitiongraphc p (D) fori=1,2,...,n 3. Since Out(v i ) = {v i,v i+1,...,v i+p } and n p+3, for Out(v i ) and Out(v j ), where j i 2 and i,j 0,n 1, the followings hold: (i) If j i = 2, Out(v j ) does not contain v i and v i+1, (ii) If j i p, Out(v j ) does not contain v i+p 1 and v i+p, (iii) If 2 < j i < p, Out(v j ) does not contain v i+1 and v i+2. Thus there exist at least two elements of Out(v j ) Out(v i ). Since Out(v j ) = p+1, Out(v i ) Out(v j ) p+1 2 = p 1 < p. So v i is not adjacent to

7 Note on p-competition Graphs and Paths 581 v j in the p-competition graph C p (D) for i,j = 1,2,...,n 2, where j i 2. Therefore C p (D) is P n. 3. A Sum Operation The sum G + H of two graphs G and H is the graph with the vertex set V(G+H) = V(G) V(H) and the edge set E(G+H) = E(G) E(H) {uv ; v V(G),u V(H)}. We obtain the following result on the sum G+K n. G u 1 u 2 H u 1 u 2 Sum G+H Figure 4: Graphs G,H and the sum G+H Proposition 3.1. Let G be a p-competition graph without isolated vertices. Then G+K n (n 1) is a (p+n)-competition graph. Proof. Let D be a digraph whose p-competition graph is G and V(D) = {,,...,v m }. We construct a digraph D as follows; (i) V(D ) = V(D) {u 1,u 2,...,u n }, (ii) A(D ) = A(D) ( i=1,2,...,m {v i u j j = 1,2,...,n}) ( i=1,2,...,n {u i v j and u i u k j = 1,2,...,m, k = 1,2,...,n}). v 5 v 4 Digraph D, Digraph D u 1 u 2 Figure 5: Digraphs D and D

8 582 Y. Kidokoro et al. Let G be a (p + n)-competition graph C p+n (D ). Since G has no isolated vertices, p m. Then i=1,2,...,n Out(u i) = m + n p + n and {u i i = 1,2,...,n} is a clique in C p+n (D ). Since G is a p-competition graph, if v i is adjacent to v j for i j, then Out(v i ) Out(v j ) = {u 1,u 2,...,u n } {w v i w and v j w in D } p+n. Thus {,,...,v m } V is G in the (p+n)-competition graph G. Since Out(v i ) Out(u j ) = {u 1,u 2,...,u n } {v j v i v j in D } for i = 1,2,...,m and j = 1,2,...,n, Out(v i ) Out(u j ) p+n and v i is adjacent to u j in G. Therefore C p+n (D ) = G+K n. By Proposition 3.1 we obtain the next result. Corollary 3.2. Let G be a p-competition graph without isolated vertices. Then G+K n (n 1) is also a p-competition graph. Proof. Let D be a digraph whose p-competition graph is G and V(D) = {,,...,v m }. We construct a digraph D as follows: (i) V(D ) = V(D) {u 1,u 2,...,u n }, (ii) A(D ) = A(D) i=1,2,...,n {u i v j j = 1,2,...,m}. Let G be the p-competition graph C p (D ) of D. Then for i = 1,2,...,n, Out(u i ) = {,,...,v m } and p m. Thus {u 1,u 2,...,u n } is a clique of G. Since G is a p-competition graph, {,,...,v m } V is G. For i = 1,2,...,m, Out(v i ) p and Out(v i ) {,,...,v m }. Thus v i is adjacent to u j for i = 1,2,...,m and j = 1,2,...,n. Therefore C p (D ) = G+K n. 4. Wheels A wheel W n is the graph C n 1 + K 1. Let v 0 be a degree n 1 vertex of W n and v i (i = 1,2,...,n 1) be degree three vertices of W n. And E(W n ) = {{, },{, },...,{v n 1, },{v 0, },{v 0, },...,{v 0,v n 1 }}. We have the next result on wheels. Proposition 4.1. Let W n be a wheel with order n and p be a positive integer. Then W n is a p-competition graph if and only if n p+3. Proof. We assume that W n is a p-competition graph and n p + 2. Let D be a digraph whose p-competition graph is W n, V(D) = {v 0,,...,v n 1 } and F = {In(v 0 ),In( ),...,In(v n 1 )}. Since C p (D) is W n, for an edge {v i,v j } in W n, Out(v i ) Out(v j ) p. Thus there exist at least p sets in F which contain v i and v j. So for any edge v i v j E(W n ), there exists the set k J In(v k) which contains v i and v j, where J ( ) p. For each vertex pair u and w of k J In(v k), where J

9 Note on p-competition Graphs and Paths 583 v 0 v 4 Wheel W 5 Figure 6: A wheel W 5 ( ) p, Out(u) Out(w) contains {vk k J} and Out(u) Out(w) p. Thus k J In(v k) is a clique of W n. Therefore { k J In(v k) J ( ) p } is an edge clique cover of W n and F = {In(v 0 ),In( ),...,In(v n 1 )} is a p-edge clique cover of W n. We assume that for i 0, v i V(W n ) belongs to all In(v k ) F. Since v i+1 and v i+2 are adjacent in W n, there exists the set k J In(v k) which contains v i+1 and v i+2, where J ( ) p. Then this set k J In(v k) contains v i. So v i is adjacent to v i+2, which is a contradiction. Thus v i belongs to at most n 1 sets of F. Since v i and v i+1 are adjacent in W n, and v i 1 and v i are adjacent in W n, there exists the set k J In(v k) which contains v i and v i+1, and the set k K In(v k) which contains v i 1 and v i, where J,K ( ) p. On the other hand, since v i 1 are not adjacent to v i+1, the number of sets of F which contain v i 1 and v i+1 is at most p 1. Thus the number of sets of F which contain v i is at least (p + p) (p 1) = p + 1. So we have that p+1 {In(v k ) F v i In(v k )} n 1. Since p n 2, p+1 n 1 and {In(v k ) F v i In(v k )} =n 1. Wealsoobtain {In(v k ) F v i+1 In(v k )} = n 1 and {In(v k ) F v i+2 In(v k )} = n 1. Since v i+1 is adjacent to v i and v i+2, v i is not adjacent to v i+2. Thus {v i,v i+1 } belongs to at most n 2 common sets in F. So p = n 2, because p n 2. Since F is n, {v i,v i+2 } belongs to at least n 2 common sets in F. Thus v i and v i+2 are adjacent, which is a contradiction. Therefore n p+3. By Theorem 1.5, C n is a (p 1)-competition graph, because n p+3 and n 1 (p 1) + 3. Thus W n = Cn 1 + K 1 is a p-competition graph by Proposition 3.1. References [1] C.A. Anderson, L. Langley, J.R. Lundgren, P.A. McKenna, S.K. Merz, New classes of p-competition graphs and φ-tolerance competition graphs, Congressus

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