Geňa Hahn

Graph homomorphisms: structure and symmetry

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex-transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.

A note on k-cop, l-robber games on graphs

We give an algorithmic characterisation of finite cop-win digraphs. The case of k>1 cops and k>=l>=1 robbers is then reduced to the one cop case. Similar characterisations are also possible in many situations where the movements of the cops and/or the robbers are somehow restricted.

Cycles and rays

Linkability in Countable-Like Webs.- Decomposition into Cycles I: Hamilton Decompositions.- An Order- and Graph- Theoretical Characterisation of Weakly Compact Cardinals.- Small Cycle Double Covers of Graphs.- ?-Transformations, Local Complementations and Switching.- Two Extremal Problems in Infinite Ordered Sets and Graphs: Infinite Versions of Menger and Gallai-Milgram Theorems for Ordered Sets and Graphs.- Chvatal-Erd?s Theorem for Digraphs.- Long Cycles and the Codiameter of a Graph II.- Compatible Euler Tours in Eulerian Digraphs.- A.J.W. Hilton, C.A. Rodger, Edge-Colouring Graphs and Embedding Partial Triple Systems of Even Index.- On the Rank of Fixed Point Sets of Automorphisms of Free Groups.- On Transition Polynomials of 4-Regular Graphs.- On Infinite n-Connected Graphs.- Ordered Graphs Without Infinite Paths.- Ends of Infinite Graphs, Potential Theory and Electrical Networks.- Topological Aspects of Infinite Graphs.- Dendroids, End-Separators, and Almost Circuit-Connected Trees.- Partition Theorems for Graphs Respecting the Chromatic Number.- Vertex-Transitive Graphs That Are Not Cayley Graphs.

Some bounds on the injective chromatic number of graphs

A k-coloring of a graph G=(V,E) is a mapping c:V→{1,2,…,k}. The coloring c is injective if, for every vertex v∈V, all the neighbors of v are assigned with distinct colors. The injective chromatic number χi(G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K4-minor free graph G with maximum degree Δ≥1 has \(\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil\). Moreover, some related results and open problems are given.

On the ultimate independence ratio of a graph

Abstract We study the ultimate independence ratio I(G) of a graph G, defined as the limit, as k→∞, of the sequence of independence ratios of the powers Gk. We prove that I(H)≤I(G) if there is a homomorphism of G to H. This allows us to prove that 1/χ(G) ≤ I(G) ≤ 1/χ ƒ (G) , where χ(G) and χ ƒ (G) are the chromatic and the fractional chromatic numbers of G, respectively. From this result we derive a number of consequences: we construct graphs with I(G) strictly between 1/χ(G) and i(G) (answering a question from an earlier paper). We estimate I(G) for some classes of graphs and, in some cases, we compute the exact value of I(G). In particular, we show that I(G) = 1 2 for all bipartite graphs G.

Absorbing sets in arc-coloured tournaments

Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of T absorbing if from each vertex of T not in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches. The greater part of the paper deals with finite tournaments, the last section treats infinite ones. In each case questions are suggested, both old and new.

On cop-win graphs☆

Abstract Following a question of Anstee and Farber we investigate the possibility that all bridged graphs are cop-win. We show that infinite chordal graphs, even of diameter two, need not be cop-win and point to some interesting questions, some of which we answer.

Some results on the injective chromatic number of graphs

A k-coloring of a graph G=(V,E) is a mapping c:V→{1,2,…,k}. The coloring c is injective if, for every vertex v∈V, all the neighbors of v are assigned with distinct colors. The injective chromatic number χi(G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K4-minor free graph G with maximum degree Δ≥1 has

Edge-Ends in Countable Graphs

\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil

Counting feasible solutions of the traveling salesman problem with pickups and deliveries is #P-complete

. Moreover, some related results and open problems are given.