A unit disk graph is the intersection graph of a family of unit disks in the plane. If the disks do not overlap, it is also a unit coin graph or penny graph. It is known that finding a maximum independent set in a unit disk graph is a NP-hard problem. In this work we extend this result to penny graphs. Furthermore, we prove that finding a minimum clique partition in a penny graph is also NP-hard, and present two linear-time approximation algorithms for the computation of clique partitions: a 3-approximation...

A unit disk graph is the intersection graph of a family of unit disks in the plane. If the disks do not overlap, it is also a unit coin graph or penny graph. It is known that finding a maximum independent set in a unit disk graph is a NP-hard problem. In this work we extend this result to penny graphs. Furthermore, we prove that finding a minimum clique partition in a penny graph is also NP-hard, and present two linear-time approximation algorithms for the computation of clique partitions: a 3-approximation...

If G is a minimally 3-connected graph and C is a double cover of the set of edges of G by irreducible walks, then |E(G)| ≥ 2| C| - 2.

In this paper, we establish a theorem on Möbius inversion over power set lattices which strongly generalizes an early result of Whitney on graph colouring.

A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . . , k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.

A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is...

A $k$-ranking of a graph $G=(V,E)$ is a mapping $\varphi V\to \{1,2,\cdots ,k\}$ such that each path with endvertices of the same colour $c$ contains an internal vertex with colour greater than $c$. The ranking number of a graph $G$ is the smallest positive integer $k$ admitting a $k$-ranking of $G$. In the on-line version of the problem, the vertices $v_1,v_2,\cdots ,v_n$ of $G$ arrive one by one in an arbitrary order, and only the edges of the induced graph $G\left[\{v_1,v_2,\cdots ,v_i\}\right]$ are known when the colour for the vertex $v_i$ has to be chosen. The on-line ranking number of a graph $G$ is the smallest...

A 2-packing of a hypergraph 𝓗 is a permutation σ on V(𝓗) such that if an edge e belongs to 𝓔(𝓗), then σ (e) does not belong to 𝓔(𝓗). We prove that a hypergraph which does not contain neither empty edge ∅ nor complete edge V(𝓗) and has at most 1/2n edges is 2-packable. A 1-uniform hypergraph of order n with more than 1/2n edges shows that this result cannot be improved by increasing the size of 𝓗.

We study domination between different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks). We succeeded in characterizing those graphs in which every uv-walk of one particular kind dominates every uv-walk of other specific kind. We thereby obtained new characterizations of standard graph classes like chordal, interval and superfragile graphs.

For an integer $k\ge 2$ and a $k$-uniform hypergraph $H$, let ${\delta }_{k-1}\left(H\right)$ be the largest integer $d$ such that every $(k-1)$-element set of vertices of $H$ belongs to at least $d$ edges of $H$. Further, let $t(k,n)$ be the smallest integer $t$ such that every $k$-uniform hypergraph on $n$ vertices and with ${\delta }_{k-1}\left(H\right)\ge t$ contains a perfect matching. The parameter $t(k,n)$ has been completely determined for all $k$ and large $n$ divisible by $k$ by Rödl, Ruci’nski, and Szemerédi in [Perfect matchings in large uniform hypergraphs with large minimum collective degree, submitted]....

The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_f$ of all finite undirected graphs. If G is a graph from $C_f$, then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

A graph is called perfect matching compact (briefly, PM-compact), if its perfect matching graph is complete. Matching-covered PM-compact bipartite graphs have been characterized. In this paper, we show that any PM-compact bipartite graph G with δ (G) ≥ 2 has an ear decomposition such that each graph in the decomposition sequence is also PM-compact, which implies that G is matching-covered