Jan Ekstein

The rainbow connection number of 2-connected graphs

Abstract The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al., we prove that the rainbow connection number of every 2-connected graph with n vertices is at most ⌈ n / 2 ⌉ . The bound is optimal.

Packing chromatic number of distance graphs

The packing chromatic number ? ? ( G ) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X 1 , ? , X k where vertices in X i have pairwise distance greater than i . We study the packing chromatic number of infinite distance graphs G ( Z , D ) , i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i , j ? Z are adjacent if and only if | i - j | ? D .In this paper we focus on distance graphs with D = { 1 , t } . We improve some results of Togni who initiated the study. It is shown that ? ? ( G ( Z , D ) ) ? 35 for sufficiently large odd t and ? ? ( G ( Z , D ) ) ? 56 for sufficiently large even t . We also give a lower bound 12 for t ? 9 and tighten several gaps for ? ? ( G ( Z , D ) ) with small t .

Radio labelings of distance graphs

Motivated by the Channel Assignment Problem, we study radio k-labelings of graphs. A radio k-labeling of a connected graph G is an assignment c of non-negative integers to the vertices of G such that |c(x)-c(y)|>=k+1-d(x,y), for any two vertices x and y, x y, where d(x,y) is the distance between x and y in G. In this paper, we study radio k-labelings of distance graphs, i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i,j@?Z are adjacent if and only if |i-j|@?D. We give some lower and upper bounds for radio k-labelings of distance graphs with distance sets D(1,2,...,t), D(1,t) and D(t-1,t) for any positive integer t>1.

3-Coloring Triangle-Free Planar Graphs with a Precolored 9-Cycle

Given a triangle-free planar graph G and a cycle C of length 9 in G, we characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results for the length of C up to 8.

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common vertex. It is not known whether every 3 longest paths in a connected graph have a common vertex and similarly for 4, 5, and 6 longest path. In [5] the authors give an upper bound on distance among 3 longest paths in a connected graph. In this paper we give a similar upper bound on distance between 4 longest paths and also for k longest paths, in general.