Jerzy Michael

Degree sequences of highly irregular graphs

We call a simple graph highly irregular if each of its vertices is adjacent only to vertices with distinct degrees. In this paper we examine the degree sequences of highly irregular graphs. We give necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a highly irregular graph.

Highly irregular graphs with extreme numbers of edges

Abstract A simple connected graph is highly irregular if each of its vertices is adjacent only to vertices with distinct degrees. In this paper we find: (1) the greatest number of edges of a highly irregular graph with n vertices, where n is an odd integer (for n even this number is given in [1]), (2) the smallest number of edges of a highly irregular graph of given order.

A smallest irregular oriented graph containing a given diregular one

A digraph is called irregular if its vertices have mutually distinct ordered pairs of semi-degrees. Let D be any diregular oriented graph (without loops or 2-dicycles). A smallest irregular oriented graph F, F=F(D), is constructed such that F includes D as an induced subdigraph, the smallest digraph being one with smallest possible order and with smallest possible size. If the digraph D is arcless then V(D) is an independent set of F(D) comprising almost all vertices of F(D) as |V(D)|->~. The number of irregular oriented graphs is proved to be superexponential in their order. We could not show that almost all oriented graphs are/are not irregular.

The minimum size of fully irregular oriented graphs

Abstract Digraphs in which any two vertices have different pairs of semi-degrees are called fully irregular. For n-vertex fully irregular oriented graphs (i.e. digraphs without loops or 2-dicycles) the minimum size is presented.

Degree sequences of digraphs with highly irregular property

A digraph such that for each its vertex, vertices of the outneighbourhood have different in-degrees and vertices of the inneighbourhood have different out-degrees, will be called an HI-digraph. In this paper, we give a characterization of sequences of pairs of outand in-degrees of HI-digraphs.

Some properties of vertex-oblique graphs

The type t G ( v ) of a vertex v ? V ( G ) is the ordered degree-sequence ( d 1 , ? , d d G ( v ) ) of the vertices adjacent with v , where d 1 ? ? ? d d G ( v ) . A graph G is called vertex-oblique if it contains no two vertices of the same type. In this paper we show that for reals a , b the class of vertex-oblique graphs G for which | E ( G ) | ? a | V ( G ) | + b holds is finite when a ? 1 and infinite when a ? 2 . Apart from one missing interval, it solves the following problem posed by Schreyer et?al. (2007): How many graphs of bounded average degree are vertex-oblique? Furthermore we obtain the tight upper bound on the independence and clique numbers of vertex-oblique graphs as a function of the number of vertices. In addition we prove that the lexicographic product of two graphs is vertex-oblique if and only if both of its factors are vertex-oblique.

A Sokoban-type game and arc deletion within irregular digraphs of all sizes

Digraphs in which ordered pairs of outand in-degrees of vertices are mutually distinct are called irregular, see Gargano et al. [3]. Our investigations focus on the problem: what are possible sizes of irregular digraphs (oriented graphs) for a given order n? We show that those sizes in both cases make up integer intervals. The extremal sizes (the 612 Z. Dziechcińska-Halamoda, Z. Majcher, ... endpoints of these intervals) are found in [1, 5]. In this paper we construct, with help of Sokoban-type game, n-vertex irregular oriented graphs (irregular digraphs) of all intermediate sizes.

Extremum degree sets of irregular oriented graphs and pseudodigraphs

A digraph in which any two vertices have distinct degree pairs is called irregular. Sets of degree pairs for all irregular oriented graphs 318 Z. Dziechcińska-Halamoda, Z. Majcher, J. Michael and ... (also loopless digraphs and pseudodigraphs) with minimum and maximum size are determined. Moreover, a method of constructing corresponding irregular realizations of those sets is given.