For a 2-connected graph $G$ on $n$ vertices and two $x,y\in V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if are $\frac{n-1}{2}$ in $V(G)\backslash \{x,y\}$ degree $k$. This strengthens well-known theorem due to Erd\H{o}s Gallai 1959. As the first application this result, show with contains cycle $2k$ it has $\frac{n}{2}+k$ confirms 1975 conjecture made by Woodall. anothe...

Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343–358) introduced a graph operator D(G), called the D-graph of G, which has been useful in investigating the structural aspects of maximal Tutte sets in G with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph D(G) and maximal extreme sets in G. Th...

Let G be a connected graph with maximum degree ∆. Brooks’ theorem states that G has a ∆-coloring unless G is a complete graph or an odd cycle. A graph G is degree-choosable if G can be properly colored from its lists whenever each vertex v gets a list of d(v) colors. In the context of list coloring, Brooks’ theorem can be strengthened to the following. Every connected graph G is degree-choosabl...

We consider the problem of orienting the edges of a graph so that the length of a longest path in the resulting digraph is minimum. As shown by Gallai, Roy and Vitaver, this edge orienting problem is equivalent to finding the chromatic number of a graph. We study various properties of edge orienting methods in the context of local search for graph coloring. We then exploit these properties to d...

Havel in 1955 [28], Erdős and Gallai in 1960 [21], Hakimi in 1962 [26], Ruskey, Cohen, Eades and Scott in 1994 [69], Barnes and Savage in 1997 [6], Kohnert in 2004 [49], Tripathi, Venugopalan and West in 2010 [83] proposed a method to decide, whether a sequence of nonnegative integers can be the degree sequence of a simple graph. The running time of their algorithms is Ω(n) in worst case. In th...

This paper examines the structure of largest subgraphs Erd?s–Rényi random graph, G n , p with a given matching number. extends result Erd?s and Gallai who, in 1959, gave classification structures K We show that their to high probability when ? 8 ln or ? 1 but it does not extend (again probability) 4 ( 2 e ) < 3 .

For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ minimum number colors needed for have q)$-coloring. This was introduced about 45 years ago, but studied systematically by Erd\H{o}s Gy\'{a}rf\'{a}s in 1997, now known as Erd\H{o}s-Gy\'{a}rf\'{a}s function. In this p...

Havel in 1955 [28], Erdős and Gallai in 1960 [21], Hakimi in 1962 [26], Ruskey, Cohen, Eades and Scott in 1994 [69], Barnes and Savage in 1997 [6], Kohnert in 2004 [49], Tripathi, Venugopalan and West in 2010 [83] proposed a method to decide, whether a sequence of nonnegative integers can be the degree sequence of a simple graph. The running time of their algorithms is Ω(n) in worst case. In th...

Youngs, D.A., Gallai’s problem on Dirac’s construction, Discrete Mathematics 101 (1992) 343-350. It is thought that T. Gallai posed the following problem concerning a construction due to G.A. Dirac: suppose that a graph K consists of disjoint subgraphs G and Hand a set of edges joining them. If each of G, H, and K are colour critical graphs, under what circumstances is it then true that the joi...