This note concerns the f -parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with degF ≤ f and minimizing the number of vertices x with degF (x) ≡ f(x) mod 2. First we prove a Gallai–Edmonds type structure theorem and some other known results on the f -parity subgraph problem, ...

We present a short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem based on Ore’s Defect Formula and Anderson’s proof of Tutte’s 1-Factor Theorem from Hall’s Theorem. The fundamental theorems on matchings in graphs have been proved in many ways, and generally they all imply each other. The most well-known is Hall’s Theorem [7] characterizing when a bipartite graph has ...

Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erdős-Rényi model, such as dependent edges and non-binomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows su...

Zverovich, I.E. and V.E. Zverovich, Contributions to the theory of graphic sequences, Discrete Mathematics 105 (1992) 293-303. In this article we present a new version of the ErdGs-Gallai theorem concerning graphicness of the degree sequences. The best conditions of all known on the reduction of the number of Erdiis-Gallai inequalities are given. Moreover, we prove a criterion of the bipartite ...

4 History of the Sylvester-Gallai Problem 6 4.1 Melchior’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Motzkin’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.3 Kelly and Moser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.4 Green and Tao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.5 How to describe 3-color ...

ly, the algorithm works as follows: • Start with a complete undirected graph on all variables. • For each pair of variables, see if conditioning on some set of variables makes them conditionally independent; if so, remove their edge. • Identify all colliders by checking for conditional dependence; orient the edges of colliders. • Try to orient undirected edges by consistency with already-orient...

A finite sequence of nonnegative integers is called graphic if the terms in the sequence can be realized as the degrees of vertices of a finite simple graph. We present two new characterizations of graphic sequences. The first of these is similar to a result of HavelHakimi, and the second equivalent to a result of Erdős & Gallai, thus providing a short proof of the latter result. We also show h...

In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K(p(1 ) ..... p(n)), conditions that are reminiscent of the Erd6s-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this result to investigate the maximum number o...

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In a 2003 paper, Kawarabayashi, Plummer and Saito showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only...

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.