Ewa Drgas-Burchardt

GENERALIZED LIST COLOURINGS OF GRAPHS

1. Introduction and NotationWe consider finite undirected graphs without loops and multiple edges. Thevertex set of a graph G is denoted by V(G) and the edge set by E(G). Thenotation H ⊆ G means that H is a subgraph of G. The vertex induced (wewill say briefly: induced) subgraph H of G is denoted by H ≤G. We say thatG contains H whenever G contains a subgraph isomorphic to H. In general,we follow the notation and terminology of [5].Let I denote the set of all mutually nonisomorphic graphs. If P is anonempty subset of I, then P will also denote the property that a graph is

On chromatic polynomials of hypergraphs

Abstract We consider a natural generalization of the chromatic polynomial of a graph. Let f ( x 1 , … , x m ) ( H , λ ) denote a number of different λ-colourings of a hypergraph H = ( X , E ) , X = { v 1 , … , v n } , E = { e 1 , … e m } , satisfying that in an edge e i it is used at least x i different colours. In the paper we show that f ( x 1 , … , x m ) ( H , λ ) can be expressed by a polynomial in λ of degree n and as a sum of graph chromatic polynomials. Moreover, we present a reduction formula for calculating f ( x 1 , … , x m ) ( H , λ ) . It generalizes the similar formulas observed by H. Whitney and R.P. Jones for standard colourings of graphs and hypergraphs respectively. We also study some coefficients of f ( x 1 , … , x m ) ( H , λ ) and their connection with the sizes of the edges of H .

Classes of chromatically unique graphs

We prove that graphs obtained from complete equibipartite graphs by deleting some independent sets of edges are chromatically unique.

A number of stable matchings in models of the Gale-Shapley type

We give some upper bounds on the maximum number of stable matchings in the Gale-Shapley marriage model with n men and n women. We also characterize, with the use of some graph-theoretical notions, the exact number of such matchings, assuming that the preferences of men and women are given.

Harmonious colourings of graphs

Abstract A λ -harmonious colouring of a graph G is a mapping from V ( G ) into { 1 , … , λ } that assigns colours to the vertices of G such that each vertex has exactly one colour, adjacent vertices have different colours, and any two edges have different colour pairs. The harmonious chromatic number h ( G ) of a graph G is the least positive integer λ , such that there exists a λ -harmonious colouring of G . Let h ( G , λ ) denote the number of all λ -harmonious colourings of G . In this paper we analyse the expression h ( G , λ ) as a function of a variable λ . We observe that this is a polynomial in λ of degree ∣ V ( G ) ∣ , with a zero constant term. Moreover, we present a reduction formula for calculating h ( G , λ ) . Using reducing steps we show the meaning of some coefficients of h ( G , λ ) and prove the Nordhaus–Gaddum type theorem, which states that for a graph G with diameter greater than two h ( G ) + 1 2 χ ( G 2 ¯ ) ≤ ∣ V ( G ) ∣ , where χ ( G 2 ¯ ) is the chromatic number of the complement of the square of a graph G . Also the notion of harmonious uniqueness is discussed.

Classes of chromatically unique or equivalent graphs

Abstract We prove the chromatic uniqueness of some families of graphs by using a special function which is the difference between the size of the line graph and the number of triangles of a given graph. Moreover, this function can be helpful in finding graphs chromatically equivalent to a given graph. We give a few examples showing how to generate such classes of graphs by using this function.

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.

The deficiency of all generalized Hertz graphs and minimal consecutively non-colourable graphs in this class

A?proper edge colouring of a?graph with natural numbers is consecutive if colours of edges incident with each vertex form an?interval of integers. The?deficiency def ( G ) of a?graph G is the?minimum number of pendant edges whose attachment to G makes it consecutively colourable. In 1999 Giaro, Kubale and Malafiejski considered the?deficiency of the?Hertz graphs. In this paper we study the?deficiency of graphs from much wider class, which we call the?generalized Hertz graphs. We find the?exact values of the?deficiency of all graphs from this class. Our investigation confirms, in this class, the?conjecture that the?deficiency of an?arbitrary graph is not greater than its order. Moreover, we describe necessary and sufficient conditions which guarantee that the?generalized Hertz graph is consecutively colourable, and necessary and sufficient conditions which guarantee that such a graph is minimal consecutively non-colourable. Applying the?last mentioned result, we give the?generating function for the?sequence whose specified elements represent numbers of the?minimal generalized Hertz graphs that are not consecutively colourable. One can find (Petrosyan and Khachatrian, 2014) the sufficient condition for consecutive non-colourability of bipartite graphs constructed based on trees in which any two leaves are in an even distance. In the paper we show that the same condition is also necessary for generalized Hertz graphs.

On the structure and deficiency of k -trees with bounded degree

A?proper edge colouring of a?graph with natural numbers is consecutive if colours of edges incident with each vertex form a?consecutive interval of integers. The?deficiency d e f ( G ) of a?graph G is the?minimum number of pendant edges whose attachment to G makes it consecutively colourable. Since all 1-trees are consecutively colourable, in this paper we study the?deficiency of k -trees for k ? 2 . Our investigation establishes the?values of the?deficiency of all k -trees that have maximum degree bounded from above by 2 k , with k ? { 2 , 3 , 4 } . To obtain these results we consider the?structure of k -trees with bounded degree and the?deficiency of general graphs of odd order. In the?first case we show that for n ? 2 k + 3 the?structure of an? n -vertex k -tree with maximum degree not greater than 2 k is unique. In the second one we prove that for each n -vertex graph G of odd order the?inequality d e f ( G ) ? 1 2 ( | E ( G ) | - ( n - 1 ) Δ ( G ) ) holds. Both last mentioned results seem to be interesting in their own right.

A Note on the Uniqueness of Stable Marriage Matching

Abstract In this note we present some sufficient conditions for the uniqueness of a stable matching in the Gale-Shapley marriage classical model of even size. We also state the result on the existence of exactly two stable matchings in the marriage problem of odd size with the same conditions.