A note on a theorem of Erdös & Gallai

A remark concerning graphical sequences

In “A note on a theorem of Erdös & Gallai” ([6]) one identifies the nonredundant inequalities in a characterization of graphical sequences. We explain how this result may be obtained directly from a simple geometrical observation involving weak majorization. A sequence of positive integers d1, d2, . . . , dp is called graphical if it is the degree sequence of a graph, i.e., there is a graph who...

Sylvester-Gallai Theorem and Metric Betweenness

Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with li...

Note on the Converse of Fabry's Gap Theorem

A survey of the studies on Gallai and anti-Gallai graphs

The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph L(G). The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in L(G). Attributed to Gallai (1967), the study ...

A note on spectral mapping theorem

This paper aims to present the well-known spectral mapping theorem for multi-variable functions.