Our goal is to extend the following result of Erd˝ os and Gallai for hypergraphs: Theorem 1 (Erd˝ os-Gallai [1]) Let G be a graph on n vertices containing no path of length k. Then e(G) ≤ 1 2 (k − 1)n. Equality holds iff G is the disjoint union of complete graphs on k vertices. We consider several generalizations of this theorem for hypergraphs. This is due to the fact that there are several po...

This text together with the attached paper [8] surveys results on color-critical graphs, with emphasis on sparse ones. The first two sections discuss the important contributions by Dirac and Gallai and present proofs of some remarkable results of them. The next two sections discuss the later progress and a number of applications of the recent results. We also use [8] for description of some app...

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to its connection to maximum matchings in a graph. In this paper, we first define θ-barrier sets. Our definition of a θ-barrier set is slightly different from that of a barrier set. However we show that θ-barrier sets and barrier sets have similar properties. In particular, we prove a generalized Berge...

A graph G is called k-critical if G is k-chromatic but every proper subgraph of G has chromatic number at most k 1. In this paper the following result is proved. If G is a k-critical graph (k>~4) on n vertices, then 21E(G)I>(k 1)n ÷ ((k 3)/(k 2 3))n + k 4 where n>~k + 2 and n ~ 2 k 1. This improves earlier bounds established by Dirac (1957) and Gallai (1963). (~) 1998 Elsevier Science B.V. All ...

Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O(n) cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph G(n, p) with probability approaching 1 as n → ∞. In this paper we show that for most edge probabilities G(n, p) can be decomposed into a union of n4 + np 2 + o(n...

All graphs considered here are directed and have no loops. For a directed graph D, let V(D) denote the set of vertices of D, 2E(D) its set of arcs, and z(D) the chromatic number of D. D is symmetric iff (x, y)~E(D)~(y, x)EE(D). A directed walk of length k in D is a sequence o f k arcs (not necessarily distinct), e~, e2, ..., ek such that the initial vertex of ei+l is the terminal vertex of e, f...

In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path-decomposition of G having at most dn2 e paths. In this paper, we show that for any simple graph G having girth g ≥ 4, there is a path-decomposition of G having at most p(G) 2 + ⌊( g+1 2g ) q(G) ⌋ paths, where p(G) is the number of vertices of odd degree in G and q(G) is the number of non-isolat...

Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove that every connected graph of even order has a vertex partition into sets inducing subgraphs with all degrees odd, and give bounds for the number of sets of this type required for vertex partitions and vertex covers. We also give results on the partitioning a...

Tutte introduced a V by V skew-symmetric matrix T = (tij), called the Tutte matrix, associated with a simple graph G= (V,E). He associates an indeterminate ze with each e ∈ E, then defines tij = ±ze when ij = e ∈ E, and tij = 0 otherwise. The rank of the Tutte matrix is exactly twice the size of a maximum matching of G. Using linear algebra and ideas from the Gallai–Edmonds decomposition, we de...

A graph G is called k-critical if it has chromatic number k; but every proper subgraph of G is ðk 1Þ-colourable. We prove that every k-critical graph ðkX6Þ on nXk þ 2 vertices has at least 1 2 ðk 1þ k 3 ðk cÞðk 1Þþk 3Þn edges where c 1⁄4 ðk 5Þð2 1 ðk 1Þðk 2ÞÞ: This improves earlier bounds established by Gallai (Acad. Sci. 8 (1963) 165) and by Krivelevich (Combinatorica 17 (1999) 401). r 2002 El...