The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we propose a family of representations for the graph G, which yield the well-known upper bound [1)] , where d is the maximum degree of G. The extremal graphs for even d are also described, and the upper bound on the interval number in terms of the number of edges of G is improved

Pluhár András

English

University of Szeged

Acta Cybernetica 12 (1995) 125-129. Remarks on the Interval Number of Graphs A. Pluhár* Abstract The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we propose a family of representations for the graph G, which yield the well-known upper bound [ 1 ) ] , where d is the maximum degree of G. The extremal graphs for even d are also described, and the upper bound on the interval number in terms of the number of edges of G is improved. 1 Introduction and Results It is a very natural idea to represent a graph G as the intersection graph of some sets. That is, we assign a set to each vertex of G so that v is adjacent to w if and only if the common part of the assigned sets is not empty. At — interval representation is an assignment, where each set consists of at most t closed intervals. The interval number of G, denoted by i(G), is the least integer t for which a ¿-representation of G exists. Finally, a representation is displayed if each set of the representation has an open interval disjoint from the other sets. Such an interval is called displayed segment. There are a number of published results concerning bounds on i(G), as well as applications of the interval representations [1-8]. Since for the complete graph Kn (on n vertices) i(G) = 1, the main interest lies in finding upper bounds in terms of the maximum degree, the number of vertices and the number of edges of a graph G, see in [2], [3], [6] and [8]. Theorem 1 (3) If G is a graph with maximum degree d, then i(G) < \^(d+ 1)]. The bound of Theorem 1 is sharp, since the equality is attained for example a (¿-regular, triangle-free graphs G. We shall give a new proof of Theorem 1, which is also useful in investigating the extremal graphs of the degree bound. Theorem 2 If a graph G has no d-regular, triangle-free component, then i(G) < 'Department of Computer Science, Attila József University, Árpád tér 2, H-6720 Szeged, Hungary 125 126 A. Pluhar That is to say, in the case d = 2k the extremal graphs are just the d-regular, triangle-free graphs. Unfortunately, one cannot expect to get such a simple result when d = 2k + 1. For example the graph which arise from Ki,3 subdividing all its edges [7], or Cn , n > 5 with a chord have interval number 2 with d = 3. It is possible to bound i(G) in terms of e, where e is the number of edges in G. It was conjectured in [3] that i(G) < \\\Je 1 + 1, which would be best possible because of the graphs Kim,2m for m e N. The best published result is in [6], stating that i{G) < ix/f 1 + 1- We shall improve on the estimations used in [6], and show Theorem 3 Every graph with e edges has a displayed interval representation with at most 1 -I- ] intervals for each vertex. It is necessary to state one more earlier result in order to prove Theorem 3. Theorem 4 (2) If a graph G has n > 1 vertices, then i(G) < \\(n+ 1)], and this bound is the best possible. 2 Proofs Proof of Theorem 1 We shall construct a displayed representation for the graph G such that at most + 1)1 intervals are assigned to each vertex v, where d(v) designates the degree of the vertex v. A walk W in G is just a sequence of vertices W = {v\,v2, ...,vi} such that, there is an edge between Vi and 1̂ +1 for each ¿ = 1,2,...,/ — 1. Let us partition the edges of G into minimal number of edge disjoint walks {Wi}Ji=l . Now represent the walk Wi = (uj ,v\, for 1 < i < j, assigning an Jp interval to the vertex vlp such that two intervals have intersection if and only if the corresponding vertices are next to each other in the walk Wi. This procedure leads to a displayed interval representation of G. Since a vertex v can be an endvertex of the walks at most two times, if v is represented by I intervals, then d(v) >2{l-2) + 2 = 2l-2. Hence \\(d{v) + 1)1 > \\{2l - 2 + 1)1 = \l - I ] = /. • Proof of Theorem 2 We can assume that d = 2k because of Theorem 1. Let us choose among all partitions of the edge set into a minimum number of edge disjoint walks a partition { w h i c h also minimizes the size of the set Q of vertices occuring k + 1 times in the walks {Wi}3i=1. The representation is the same as in the proof of Theorem 1. If Q = 0, we are done. For an x € Q there exists a p 6 {1, •••, j } such that x = v^, x = vp, 1 and x Wi for all I ^ p. The last statement follows from the minimality Remarks on the Interval Number of Graphs 127 of j, since in case of x — vls 6 Wi we could replace the walks Wp and Wi by the walk W* = {v[,vl2,...,vls,v%,...,vpn{p),vls+1,...,vln(l)). For any vertex y = vvs / x from Wp, we can transform the walk Wp into the walk w; = That is, by the minimality of Q, y occurs in the walks {Wi}i^p U {W*} k + 1 times. Then again, all neighbors of y are in Wp. That is the vertex set of Wp is a 2k—regular component of G. Now we can conclude the proof by showing that if a 2k—regular graph G is not triangle-free, then i{G) < k. Suppose that u, v and w span a triangle in G. If k = 1, then G = K3, and we are done. For k > 1 there is is an Euler circuit C in G, starting by v,u,w,v,x and finishing at v. But it can be represented by k intervals per vertex as in the proof of Theorem 1, just take the convex hull of the two intervals which represent v at the beginning of the walk. • Proof of Theorem 3 We need the definition of the degree sequence of a graph G first. Let us suppose that is an order of the vertices of G such that di > dj if i < j, where di = deg(vi) denotes the degree of the vertex Our argument closely follows the one in [8]. The crucial difference is the additional information about the degree sequence of G. It is gained by using Theorem 1 and an idea, which first appeared in [4]. Lemma 1 Let d\ > d2 > ... > dn be the degree sequence of a graph G. If i(G) > t + 1, then dt > 2t - i + 1. Proof of Lemma 1 Let Vi be a vertex of degree di. By Theorem 1 l\id i +1)1 >i(G) >t + l, that is £¿1 > 2t + 2. Now we partition the edges of G into directed forests, represent them one by one and remove the edges of the represented forest from G. The idea is that the representation of the Ith forest exhausts all edges adjacent to vi, and decreases the degree of all vertices in the remaining graph which still has non zero degree. The construction of the first forest F\ starts with choosing a breadth-first-search tree 2\, rooted in vi, all edges directed toward V\. If there are vertices outside of Ti, just pick arbitrary trees in which the edges are directed toward the root. The procedure for selecting Ft is similar, we take vi, the vertex of degree di as a root of a tree, and also take other trees if the remaining graph is not connected. The main point is that Ft is maximal, and all edges adjacent to vi are in U...Ui^. For the maximum degree A1 in the remaining graph Gl = G\Fi U.. .Ui^-i we have show that A i <di-{i- 1) 128 A. Pluhar by induction. On the other hand, we can represent the edges of U ... U i ) by using at most I + 1 intervals for each vertex. First assign intervals Iv to each vertex v of G such that Iv D Iw = 0 for v ^ w. Then, for each i 6 {1,..., / } if the directed edge (v,w ) is in Fi, assign a small interval to v inside in Iw, which has no common points with the other intervals. Because of Theorem 1 and the previous representation we have i(G) < i + i(G\Fi U ... U Fi_x) < i + + Since t + 1 < i(G) < i + [ d i ~ ' + 2 l , it follows that di-i + 2 t + 3 / 2 < i + - l — 2 , that is di > 2t - i + 1. • Now, with a few modifications, we may repeat the argument presented in [8]. First, partition the vertices of G into two classes, A and B. A contains the vertices of degree at least r|-\/el + 1> while the degree of a vertex from B is at most The edges between the elements of A can be represented by using at most [|(|/1| + 1)] intervals for each vertex because of Theorem 4. Let us make this system of intervals displayed by adding an isolated interval for each vertex of G in a same way as in the proof of Lemma 1. For each edges between A and B, or inside B, take an endpoint from B, and place a small interval for it into a displayed segment for its neighbor. This procedure produces at most [|\/e] + 1 intervals for an element of B. That is i(G) < max(|J>/el + 1, \\(\A\ + 1)1 + 1). In order to estimate = k, we need the identity 2e = di, where {(¿¿}"=1 is the degree sequence in decreasing order. There is nothing to prove if i(G) so we may assume that by Lemma 1. Thus if 1 k n 2e= di+ Y, di+ J2 di> i=1 i=\§\/£l + l i=k+1 which implies r l v ^ l 2 * „ 2 e > £ ( 2 f - > / S l - i + l ) + £ ( ^ 1 + 1). i=1 i= r|\/?l+i Simple computation shows that k < |ye— 1. Plugging in this estimation, one gets the bound i(G) < m a x ( r ^ l + 1, r ¿ ( f v G - 1 + 1)1 + 1) < T ^ l + 1-• Remarks on the Interval Number of Graphs 129 References [1] Andreae, T., "On the Interval Number of Triangulated Graph", J. Graph Theory 3 (1987), 273-280. [2] Griggs, J.R., "Extremal Values of the Interval Number of Graph, II.", Dis-crete Math. 28 (1979), 37-47. [3] Griggs, J.R. and West, D.B., "Extremal Values of the Interval Number of Graph, I.", SI AM J. Alg. Discrete Meth. 1 (1980), 1-8. [4] Scheinerman, E.R., "The Maximum Interval Number of Graphs with Given Genus", J. Graph Theory 3 (1987), 441-446. [5] Scheinerman, E.R. and West, D.B., "The interval number of a planar graph: three intervals suffice", Jour. Combinatorial Theory B 35 (1985), 224-239. [6] Spinrad, J.R., Vijayan, G. and West, D.B., "An Improved Edge Bound on the Interval Number of a Graph", J. Graph Theory 3 (1987), 447-449. [7] Trotter, W.T. Jr. and Harary, F. "On Double and Multiple Interval Graphs", J. Graph Theory 3 (1979), 205-211. [8] West, D.B., "A short proof of the degree bound on interval number", Discrete Math. 73 (1989), 307-310. Received May, 1995 

Remarks on the interval number of graphs

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Remarks on the interval number of graphs

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