The packing chromatic number χ ρ (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 χ ρ (G(Z, D)) ≤ 35 for sufficiently large odd t and χ ρ (G(Z, D)) ≤ 56 for sufficiently large even t. We also give a lower bound 12 for t ≥ 9 and tighten several gaps for χ ρ (G(Z, D)) with small t.
Considering connected K 1,3 -free graphs with independence number at least 3, Chudnovsky and Seymour (2010) showed that every such graph, say G, is 2ω-colourable where ω denotes the clique number of G. We study (K 1,3 , Y )-free graphs, and show that the following three statements are equivalent.(1) Every connected (K 1,3 , Y )-free graph which is distinct from an odd cycle and which has independence number at least 3 is perfect.(2) Every connected (K 1,3 , Y )-free graph which is distinct from an odd cycle and which has independence number at least 3 is ω-colourable.(3) Y is isomorphic to an induced subgraph of P 5 or Z 2 (where Z 2 is also known as hammer). * Research partly supported by the DAAD-PPP project "Colourings and connection in graphs" with project-id 57210296 (German) and 7AMB16DE001 (Czech), respectively. † Research partly supported by project P202/12/G061 of the Czech Science Foundation.Furthermore, for connected (K 1,3 , Y )-free graphs (without an assumption on the independence number), we show a similar characterisation featuring the graphs P 4 and Z 1 (where Z 1 is also known as paw).
Although it has recently been proved that the packing chromatic number is unbounded on the class of subcubic graphs, there exists subclasses in which the packing chromatic number is finite (and small). These subclasses include subcubic trees, base-3 Sierpiski graphs and hexagonal lattices. In this paper we are interested in the packing chromatic number of subcubic outerplanar graphs. We provide asymptotic bounds depending on structural properties of the outerplanar graphs and determine sharper bounds for some classes of subcubic outerplanar graphs.
The packing chromatic number χ ρ (G) of a graph G is the smallest integer p such that vertices of G can be partitioned into disjoint classes X 1 , ..., X p where vertices in X i have pairwise distance greater than i. For k < t we study the packing chromatic number of infinite distance graphs D(k, t), 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| ∈ {k, t}.We generalize results by Ekstein et al. for graphs D(1, t). For sufficiently large t we prove that χ ρ (D(k, t)) ≤ 30 for both k, t odd, and that χ ρ (D(k, t)) ≤ 56 for exactly one of k, t odd. We also give some upper and lower bounds for χ ρ (D(k, t)) with small k and t.
International audienceMotivated 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
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