This note concerns the (1, f)-odd subgraph problem, i.e. we are given an undirected graph G and an odd value function f : V (G) → N, and our goal is to find a spanning subgraph F of G with degF ≤ f minimizing the number of even degree vertices. First we prove a Gallai–Edmonds type structure theorem and some other known results on the (1, f)-odd subgraph problem, using an easy reduction to the m...

The Sylvester-Gallai theorem asserts that every finite set S of points in two-dimensional Euclidean space includes two points, a and b, such that either there is no other point in S is on the line ab, or the line ab contains all the points in S. V. Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. In the present ...

We bring together several new results related to the classical Sylvester-Gallai Theorem and its recently noted sharp dual. In 1951 Dirac and Motzkin conjectured that a configuration of n not all collinear points must admit at least n/2 ordinary connecting lines. There are two known counterexamples, when n = 7 and n = 13. We provide a construction that yields both counterexamples and show that t...

A graph G with a perfect matching is called saturated if G + e has more perfect matchings than G for any edge e that is not in G. Lovász gave a characterization of the saturated graphs called the cathedral theorem, with some applications to the enumeration problem of perfect matchings, and later Szigeti gave another proof. In this paper, we give a new proof with our preceding works which reveal...

This lecture covers the proof of the Bessy-Thomassé Theorem, formerly known as the Gallai Conjecture. Also, we discuss the cyclic stable set polytope, and show that it is totally dual integral (TDI) (see lecture 5 for more on TDI systems of inequalities). In this section we provide a brief recap of some definitions we saw in the previous lecture. Also we answer a question that remained unanswer...

Suppose G is a graph with average degree greater than k − 2. Erdős and Gallai proved that G contains a path on k vertices. In 1962, Erdős and Sós conjectured that G contains every tree on k vertices. Zhou proved the ErdősSós conjecture holds for the case where G has exactly k vertices. Wozniak proved the case where G has order k + 2. This paper’s authors proved the case where G has order k + 3 ...

If G is a triangle-free graph, then two Gallai identities can be written as α(G)+ χ(L(G)) = |V (G)| = α(L(G))+ χ(G), where α and χ denote the stability number and the clique-partition number, and L(G) is the line graph of G. We show that, surprisingly, both equalities can be preserved for any graph G by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, acc...

We take an application of the Kernel Lemma by Kostochka and Yancey [10] to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore-degree version of Brooks’ Theorem for online list-coloring. The Ore-degree of an edge xy in ...

Let G be a graph. It is well known that the maximum multiplicity of a root of the matching polynomial μ(G,x) is at most the minimum number of vertex disjoint paths needed to cover the vertex set of G. Recently, a necessary and sufficient condition for which this bound is tight was found for trees. In this paper, a similar structural characterization is proved for any graph. To accomplish this, ...