Ralph J. Faudree

Claw-free graphs—a survey

Abstract In this paper we summarize known results on claw-free graphs. The paper is subdivided into the following chapters and sections: 1. 1. Introduction 2. 2. Paths, cycles, hamiltonicity 2.1. (a) Preliminaries 2.2. (b) Degree and neighborhood conditions 2.3. (c) Local connectivity conditions 2.4. (d) Further forbidden subgraphs 2.5. (e) Invariants 2.6. (f) Squares 2.7. (g) Regular graphs 2.8. (h) Other hamiltonicity related results and generalizations 2.9. 3. Matchings and factors 2.10. 4. Independence, domination, other invariants and extremal problems 2.11. 5. Algorithmic aspects 2.12. 6. Miscellaneous 2.13. 7. Appendix — List of all 2-connected nonhamiltonian claw-free graphs on n ⩽ 12 vertices.

All Ramsey numbers for cycles in graphs

In the past, Ramsey numbers were known for pairs of cycles of lengths r and s when one of the following occurred: (1) r and s are small, (2) one of r or s is small relative to the other, or (3) r is odd and r = s. In this paper we complete the Ramsey number problem for cycles by verifying their previously conjectured values.

THE SIZE RAMSEY NUMBER

Let denote the class of all graphsG which satisfyG→(G1,G2). As a way of measuring minimality for members of, we define thesize Ramsey number ř(G1,G2) by.We then investigate various questions concerned with the asymptotic behaviour ofř.

Induced matchings in bipartite graphs

All graphs in this paper are understood to be finite, undirected, without loops or multiple edges. The graph G’ = (V’, E’) is called an induced subgraph of G = (V, E) if V’ c V and uv E E’ if and only if (u, v} c V’, uv E E. The following two problems about induced matchings have been formulated by Erdiis and NeSetiil at a seminar in Prague at the end of 1985: 1. Determine f(k, d), the maximum number of edges in a graph which has maximum degree d and contains no induced (k + Q-matching (an induced matching of k + 1 edges). For k = 1 this was asked earlier by Bermond, Bond and Peyrat (see [l]). 2. Let q*(G) denote the minimum integer t for which the edge set of G can be partitioned into t induced matchings of G. (We will call 4 *(G) the strong chromatic index of G.) As is done in Vizing’s theorem, find the best upper bound of q*(G) when G has maximum degree d. It was shown in [l] that (for d even) f(1, d) = sd2 and the extremal graph is unique (each vertex of a five cycle is multiplied by d/2). This result suggests that f(k, d) = qd2k. Perhaps a stronger conjecture is also true, namely, that q*(G) s zd2 when G has maximum degree d. In this paper the analogous extremal problem for bipartite graphs is considered. It is shown that bipartite graphs of maximum degree d without an induced (k + I)-matching have at most kd2 edges (Theorem 1). Extremal graphs for k > 1 are not unique but can be completely described (Theorem 2). It is also shown (Theorem 3) that when the extremal problem is restricted to connected bipartite graphs, the extremal number drops by at least d (if k > 2). We

Characterizing forbidden pairs for hamiltonian properties

Abstract In this paper we characterize those pairs of forbidden subgraphs sufficient to imply various hamiltonian type properties in graphs. In particular, we find all forbidden pairs sufficient, along with a minor connectivity condition, to imply a graph is traceable, hamiltonian, pancyclic, panconnected or cycle extendable. We also consider the case of hamiltonian-connected graphs and present a result concerning the pairs for such graphs.

Degree conditions for 2-factors

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ores classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k.

Neighborhood unions and hamiltonian properties in graphs

We investigate the relationship between the cardinality of the union of the neighborhoods of an arbitrary pair of nonadjacent vertices and various hamiltonian type properties in graphs. In particular, we show that if G is 2-connected, of order p ≥ 3 and if for every pair of nonadjacent vertices x and y: 1. (a) ∥N(x) ⌣ N(y)∥ ≧ (p − 1)2, then G is traceable, 2. (b) ∥N(x) ⌣ N(y)∥ ≧ (2p − 1)3, then G is hamiltonian, and if G is 3-connected and 3. (c) ∥N(x) ⌣ N(y)∥ ≧ 2p3, then G is hamiltonian-connected.

On cycle—Complete graph ramsey numbers

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K„), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices

Path Ramsey numbers in multicolorings

Abstract In this paper we consider the general Ramsey number problem for paths when the complete graph is colored with k colors. Specifically, given paths P i 1 , P i 2 ,…, P i k with i 1 , i 2 ,…, i k vertices, we determine for certain i j (1 ≤ j ≤ k ) the smallest positive integer n such that a k coloring of the complete graph K n contains, for some l , a P i l in the l th color. For k = 3, given i 2 , i 3 , the problem is solved for all but a finite number of values of i 1 . The procedure used in the proof uses an improvement of an extremal theorem for paths by P. Erdos and T. Gallai.

Weakly pancyclic graphs

The problem was posed of determining the biclique partition number of the complement of a Hamiltonian path (Monson, Rees, and Pullman, Bull. Inst. Combinatorics and Appl. 14 (1995), 17–86). We define the complement of a path P, denoted