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A Characterization of Tetrahedral Graphs*

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A Characterization of Tetrahedral Graphs* JOURNAL OF COMBINATORIAL THEORY 3, 366-385 (1967) A Characterization of Tetrahedral Graphs* R. C. BOSE AND RENU LASKAR Department of Statistics, University of North Carolina, Chapel Hill, North Carolina 27514 ABSTRACT A tetrahedral graph may be defined as a graph G, whose vertices may be identified with the n(n -- 1)(n -- 2)/6 unordered triplets on n symbols, such that two vertices are adjacent if and only if the corresponding triplets have two symbols in common. If d(x, y) denotes the distance between two vertices x and y and A(x, y) denotes the number of vertices adjacent to both x and y, then a tetrahedral graph G has the following properties: (bl) The number of vertices is n(n -- 1)(n -- 2)/6. (b2) G is connected and regular of valence 3(n 3). (ba) For any two adjacent vertices x and y, A(x, y) = n -- 2. (b4) A(x, y) = 4 if d(x, y) = 2. We show that, if n > 16, then any graph G (without loops and with utmost one edge connecting two vertices) having the properties (b0--(b~) must be a tetrahedral graph. I. INTRODUCTION 1. We shall consider only finite undirected graphs, with at most one edge joining a pair of vertices and no edge joining a vertex to itself. The valence d(u) of the vertex u of a graph G, is defined to be the number of vertices adjacent to u. If all vertices of G have the same valence nl, the graph G is said to be a regular graph of valence nl. A chain x~, x2 ..... xn is a sequence of vertices of G, not necessarily all different, such that any two consecutive vertices in the chain are adjacent. Thus the pairs (Xl, x2)(x2, xa) ..... (xn_ 1 , x~) are edges of G. The number of edges n -- 1 is said to be the length of the chain. The chain is said to begin at x~ and terminate at x~, and is said to join x~ and x~. The graph G is said to be connected if, for every pair of distinct vertices x and y, there is a chain beginning at x and terminating at y. For a con- nected graph the distance d(x, y) between two vertices x and y is defined to be the length of the shortest chain joining x and y. For any two vertices u and v, A(u, v) denotes the number of vertices w, * This research was supported by the National Science Foundation Grant No. GP- 3792 and the Air Force Office of Scientific Research Grant No. AF-AFOSR-760-65, 366 A CHARACTERIZATION OF TETRAHEDRAL GRAPHS 367 adjacent to both u and v. If u and v are adjacent, i.e., d(x, y) = 1, A(u, v) is called the edge-degree of the edge (u, v). A regular graph G for which all edges have the same edge-degree A, is said to be edge-regular, with edge- degree A. 2. A graph G is said to be triangular if the vertices of G can be identified with unordered pairs on n symbols, so that two pairs are adjacent if and only if the corresponding pairs have one symbol in common. A triangular graph G obviously possesses the following properties: (al) The number of vertices in G is n(n -- 1)/2. (a2) G is regular of valence 2(n -- 2). (a3) G is edge-regular with edge-degree n -- 2, i.e., A(u, v) = n -- 2 if u and v are adjacent. (a4) A(u, v) = 4, if u and v are non-adjacent. Connor [4] showed (with a somewhat different terminology) that, for n > 8, the properties (ai)-(a4), characterize a triangular graph, i.e.,.if G has the properties (al)-(a4), then G must be triangular. Shrikhande [9], Li-chien [7, 8] and Hoffman [5, 6] completed Connor's work by demon, strating that the same result holds for n < 8, but if n = 8, then there exist other non-isomorphic graphs which are not triangular. 3. In this paper we consider the problem of characterization of tetra- hedral graphs. A tetrahedral graph may be defined as a graph G whose Vertices can be identified with unordered triplets on n symbols, such that two vertices are adjacent if and only if the corresponding triplets have two common symbols. Itis readily seen that G has the following properties: (b0 The number of vertices in G is n(n -- 1)(n -- 2)/6. (b2) G is connected and regular of valence 3(n - 3). (b3) G is edge-regular, with edge degree n -- 2, i.e. A(x, y) = n -- 2 if d(x,y) = 1. (b4) A (x, y) = 4 if d(x, y) = 2. In Section III we prove that, for n > 16, the properties (bl)-(b4) charac- terize a tetrahedral graph, i.e., if G possesses properties (bl)-(b4) and n > 16, then it is possible to establish a (1,1) correspondence between the vertices of G, and the unordered triplets on n symbols, such that two vertices of G are adjacent if and only if the corresponding triplets have a pair of symbols in common. The proof is based on certain theorems regarding the existence or non- existence of cliques and claws in edge-regular graphs, which are proved in Section II. These theorems generalize the previous work of Bruck [2] ~and (one of the authors) Bose [1 ]. Other applications of these theorems will be given in subsequent communications. 368 BOSE AND LASKAR II. CLAWS AND CLIQUES IN EDGE-REGULAR GRAPHS 1. In this section we shall consider a graph G which has the following properties: (cl) Gis connected and regular of valence r(k -- 1). (c2) G is edge-regular with edge-degree (k -- 2) + ~. (cs) A(x, y) <~ 1 + t, for all pairs of non-adjacent vertices, x and y of G. In the above r, k, ~, t, are fixed positive integers, such that r ~ 1, k ~ 2, ~ ~ 0, fl ~ 0, and rfl -- 2~ ~ 0. All the lemmas and theorems in paragraphs 2 and 3 of this section are about the edge regular graph G, with properties (Cl), (c2), (ca). We define here some functions of the parameters r, k, ~, and t, which play an important role in subsequent developments. (2.1.1) 7(r, ~) = 1 4- (r -- 1) ~. (2.1.2) q(r, o0 = 1 4-,(2r -- l) a. (2.1.3) p(r,c~,fl) = 1 4- fl 4- (2r -- 1) o~. (2.1.4) p(r, ~, t) = 1 q- 4- 1)(rfl -- 2c0. We shall denote as usual the cardinality of a set S by I S r A clique K of a graph is a set of vertices adjacent to each other. A clique K will be called Complete if we cannot find a vertex x, not contained in K such that xUK is a clique. Thus a complete clique cannot be extended to a larger clique by the adjunction of a new vertex belonging to the graph. Now consider the graph G with the properties, (cl) , (c~), (Ca). A clique K ofG will be called a major clique if (2.1.5) I KI >/ 1 +g -- 7(r, ~) = k --(r -- 1)=. A Clique K of G will be called a grand clique if it is both major and complete. A claw [p, S] of G, consists of a vertex p, the vertex of the claw, and a non-empty set S of vertices of G, not containing p, such that p is adjacent to every vertex in S, but any two vertices in S are non-adjacent. The order of the claw is defined to be the number s = I S l In the next two paragraphs of this section, we obtain a number of theorems about claws and cliques in a graph G having the properties (Cl), (c2), (c8). These theorems are very similar to those obtained in an earlier paper [1], but are now proved under less restrictive conditions, thereby substantially increasing their range of applicability. This will be illustrated in Section III, where they will be used to obtain a geometric characterization of tetrahedral graphs (defined in Section I). Further applications will be given in subsequent communications. A CHARACTERIZATION OF TETRAHEDRAL GRAPHS 369 2. THEOREM (2.2.1). /f k > p(r,o~,fl), there cannot exist a claw of order r q- 1 in G. Suppose there exists in G a claw [p, S] of order s. Let T be the set of vertices of G, not belonging to [p, S], and adjacent to p. Let f(x) denote the number of Vertices q in T, such that q is adjacent to exactly x vertices in S. Counting the number of vertices in T we have from (c0, (2.2.1) ~f(x) = r(k-- 1) --s = rk--r--s. Counting the number of ordered pairs (b, q) where b and q are adjacent, b belongs to S, and q belongs to T, we have from (%) (2.2.2) i xf(x) = s(k -- 2 q- o0. x=O Again counting the triplets (bl, ba, q), where b I , b 2 is an ordered pair of vertices in S, q is a vertex in T, and bl b, ~ are both adjacent to q, we have from (%) (2.2.3) ~ x(x 1)f(x) ~< s(s -- 1)ft. Ifa claw of order r + 1 exists, putting s = r § 1, we have from (2.2.1), (2.2.2) and (2.2.3): 1 r+l f(O) § ~ ~ (x 1)(x -- 2)f(x) ~ --k q- 1 -t- -? 1)(rfl -- 2a) aZ=l (2.2.4) = --k + p(r, ~, 5) Since the left-hand side is essentially non-negative whereas k > p(r, o~, fl) by hypothesis, we have a contradiction.
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