Yousef Alavi

Total matchings and total coverings of graphs

In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained.

Highly irregular digraphs

A connected digraph D is highly irregular if the vertices of out-neighborhood of each vertex v?V(D) have different out-degrees. In this paper, we investigate some problems concerning the existence of highly irregular digraphs with special properties, with particular focus on highly irregular directed trees as well as their independence numbers.

On total covers of graphs

A total cover of a graph G is a subset of V(G)∪E(G) which covers all elements of V(G)∪E(G). The total covering number α2(G) of a graph G is the minimum cardinality of a total cover in G. In [1], it is proven that α2(G)⩽[n2] for a connected graph G of order n. Here we consider the extremal case and give some properties of connected graphs which have a total covering number [n2]. We prove that such a graph with even order has a 1-factor and such a graph with odd order is factor-critical.

On linear vertex-arboricity of complementary graphs

The linear vertex-arboricity ρ(G) of a graph G is defined to be the minimum number of subsets into which the vertex set of G can be partitioned such that each subset induces a linear forest. In this paper, we give the sharp upper and lower bounds for the sum and product of linear vertex-arboricities of a graph and its complement. Specifically, we prove that for any graph G of order p. and for any graph G of order p = (2n + 1)2, where n ≅ Z+, 2n + 2 ≦ ρ(G) + ρ(G).

Survey of Double Vertex Graphs

Abstract. Let G be a (V,E) graph of order p≥2. The double vertex graphU2(G) is the graph whose vertex set consists of all 2-subsets of V such that two distinct vertices {x,y} and {u,v} are adjacent if and only if |{x,y}∩{u,v}|=1 and if x=u, then y and v are adjacent in G. For this class of graphs we discuss the regularity, eulerian, hamiltonian, and bipartite properties of these graphs. A generalization of this concept is n-tuple vertex graphs, defined in a manner similar to double vertex graphs. We also review several recent results for n-tuple vertex graphs.

Highly irregular m chromatic graphs

Abstract A graph is highly irregular if it is connected and the neighbors of each vertex have distinct degrees. In this paper, we study existence and extremal problems for highly irregular graphs with a given maximum degree and focus our attention on highly irregular graphs that are m-chromatic for m⩾2.

Highly Regular Graphs

A graph G of order p 2 3 is highly regufar if there exists an n x n matrix C = [c~], where 2 5 n < p , called a collapsed adjacency matrix (CAM), such that for each vertex ZJ of G there is a partition of V(G) into n subsets Vl = {u } , V2, . . . , V,, such that each vertex y E 5 is adjacent to cij vertices in v.. This class of graphs is described in [ 1, p. 1581. In this article we develop some of the properties of highly regular graphs. We follow the notation and terminology of [3]. First we note, as examples, that the 5-cycle, the 3-cube, and the Petersen graph are highly regular. These graphs are shown in FIGURE 1, together with a collapsed adjacency matrix for each. In each graph a vertex u is selected, and every vertex is labeled with an integer i that corresponds to the subset to which the vertex belongs. A graph G is regular if every vertex of G has the same degree and is r-regular if each vertex has degree r . The term “highly regular” suggests that a graph with this property is regular. This is, in fact, the case; for suppose that G is a highly regular graph with n x n CAM C = [cij]. Let r = & cil. For any vertex z! of G, there is a partition V, = {v/, V2, V,, . . . , V, of V(G) such that 71 is adjacent to cil vertices of 6 ( i = 2, 3, . . . , n). Then deg u = r, so that G is r-regular. In general, every column sum of C i s r; for if 1 5 j 5 n a n d u E V,, then

Preface: Volume 11

Abstract The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications was held June 4-9, 2000 at Western Michigan University. This conference has been held every four years since 1968, hosted by the Department of Mathematics and Statistics at Western Michigan University. The 1996 Conference attracted nearly 300 participants from all over the world (including Canada, China, Denmark, England, France, Hungary, India, Israel, Italy, Japan, Korea, Germany, the Netherlands, Puerto Rico, Saudi Arabia, Scotland, South Africa, and Spain) and included five days of principal, invited and contributed presentations by approximately 200 speakers. Acknowledgments The excellent support of Western Michigan University, Dr. Elson S. Floyd, President Dr. Diether H. Haenicke, Past President (with his generous support of the Sixth, Seventh, Eighth, and the Ninth International Conferences) The continued support of the Division of Research and Sponsored Programs, Dr. Donald E. Thompson, Vice President The fine support of the College of Arts and Sciences, Dr. Elise Jorgens, Dean The overall support of the Department of Mathematics and Statistics, Dr. John Petro, past Chair Dr. Jay Wood, Chair The fine financial support of the Number Theory Foundation, Dr. Carl Pomerance, Director Ms. Ethel Rathburn, Secretary The special assistance of our friends and co-directors, Dr. Ronald L. Graham, Dr. John Petro and Dr. Allen J. Schwenk and associate directors Jiuqiang Liu, Michael Raines, and Ping Zhang. The support and assistance of our graph theory colleagues, Professors Arthur White and Gary Chartrand. The fine assistance of colleagues around the world who assisted with refereeing of the manuscripts. The special administrative assistance of Cheryl Peters and the fine secretarial assistance of Maryann Bovo and Margo Chapman. The dedicated work of our conference assistants, Vince Castellana, Linda Eroh, Raluca Muntean and Wendy Weaver. The editors apologize in advance for any oversight in the acknowledgments and any errors in the manuscripts. Y.A. D.R.L. J.L. D.J. Dedication For so many years and after many conferences, when leaving Kalamazoo, boarding the airplane, Paul Erdos would say “Maybe you can hold the next Conference in my memory.” Upon his passing shortly after the 8th Kalamazoo Conference, the decision to proceed with a 9th Conference in memory of Paul Erdos was easy and direct. Thus the Erdos 2000 Conference was held during June 4-9, 2000 in Kalamazoo. So it is with special devotion and honor that we dedicate these Proceedings to him. Another special friend and teacher of many colleagues here was the late Edward A. Nordhaus, to whom these special proceedings are also dedicated.

Bipartite regulation numbers

The bipartite regulation number br(G) of a bipartite graph G with maximum degree d is the minimum number of vertices required to add to G to construct a d-regular bipartite supergraph of G. It is shown that if G is a connected m-by-n bipartite graph with m ~ d - 1, then br(G) = n - m. If, however, n - m ~ 2) are integers such that 0 <~ l <~ k and 2 <~ k <~ d, then there is an connected m-by-n bipartite graph G of maximum degree d for which br(G) = n - m + 2/, for some m and n with k = d - (n - m). A well-known result of KOnig [9] states that every graph G of maximum degree d is an induced subgraph of a d-regular graph H. Erd6s and Kelly [5] determined the minimum number of vertices (the induced regulation number) which must be added to G to obtain such a supergraph H. This latter result was extended to digraphs by Beineke and Pippert [3]. The regulation number of a graph G is the minimum number of vertices which must be added to G to construct a d-regular supergraph H. In this case, G need not be an induced subgraph of H. Regulation numbers of graphs were introduced by Akiyama, Era and Harary [1], and were further studied by Akiyama and Harary [2] and Harary and Schrnidt [8]. Analogous concepts for digraphs and multigraphs were introduced by Harary and Karabed [7] and Chartrand, Harary and Oellermann [4], respectively. Here we consider the bipartite regulation numbers.

On randomly 3-axial graphs

A graph G , every vertex of which has degree at least three, is randomly 3-axial if for each vertex V of G , any ordered collection of three paths in G of length one with initial vertex V can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G . Randomly 3-axial graphs of order p > k are characterized for p ^ 1 (mod 3) , and are shown to be either complete graphs or certain regular complete bipartite graphs.