9 research outputs found
Colouring the Square of the Cartesian Product of Trees
We prove upper and lower bounds on the chromatic number of the square of the
cartesian product of trees. The bounds are equal if each tree has even maximum
degree
The Incidence Chromatic Number of Toroidal Grids
An incidence in a graph is a pair with and , such that and are incident. Two incidences and
are adjacent if , or , or the edge equals or . The
incidence chromatic number of is the smallest for which there exists a
mapping from the set of incidences of to a set of colors that assigns
distinct colors to adjacent incidences. In this paper, we prove that the
incidence chromatic number of the toroidal grid equals 5
when and 6 otherwise.Comment: 16 page
Distance two labeling of direct product of paths and cycles
Suppose that is a set of non-negative
integers and . The -labeling of graph is the function
such that if the distance
between and is one and if the
distance is two. Let and let
be the maximum value of Then is called number
of if is the least possible member of such that maintains an
labeling. In this paper, we establish numbers of graphs for all and .Comment: 13 pages, 9 figure
The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes
AbstractLet Cm be the cycle of length m. We denote the Cartesian product of n copies of Cm by G(n,m):=Cm□Cm□⋯□Cm. The k-distance chromatic number χk(G) of a graph G is χ(Gk) where Gk is the kth power of the graph G=(V,E) in which two distinct vertices are adjacent in Gk if and only if their distance in G is at most k. The k-distance chromatic number of G(n,m) is related to optimal codes over the ring of integers modulo m with minimum Lee distance k+1. In this paper, we consider χ2(G(n,m)) for n=3 and m≥3. In particular, we compute exact values of χ2(G(3,m)) for 3≤m≤8 and m=4k, and upper bounds for m=3k or m=5k, for any positive integer k. We also show that the maximal size of a code in Z63 with minimum Lee distance 3 is 26
Injective coloring of product graphs
The problem of injective coloring in graphs can be revisited through two
different approaches: coloring the two-step graphs and vertex partitioning of
graphs into open packing sets, each of which is equivalent to the injective
coloring problem itself. Taking these facts into account, we observe that the
injective coloring lies between graph coloring and domination theory.
We make use of these three points of view in this paper so as to investigate
the injective coloring of some well-known graph products. We bound the
injective chromatic number of direct and lexicographic product graphs from
below and above. In particular, we completely determine this parameter for the
direct product of two cycles. We also give a closed formula for the corona
product of two graphs
Coloring the square of the Cartesian product of two cycles
The square G 2 of a graph G is defined on the vertex set of G in such a way that distinct vertices with distance at most two in G are joined by an edge. We study the chromatic number of the square of the Cartesian product Cm✷Cn of two cycles and show that the value of this parameter is at most 7 except when m = n = 3, in which case the value is 9, and when m = n = 4 or m = 3 and n = 5, in which case the value is 8. Moreover, we conjecture that for every G = Cm✷Cn, the chromatic number of G 2 equals the size of a maximal independent set in G 2
O kromatičnem številu kvadrata kartezičnega produkta dveh ciklov
The square ▫▫ of a graph ▫▫ is obtained from ▫▫ by adding edges joining all pairs of nodes at distance 2 in ▫▫. In this note we prove that ▫ for ▫. This confirms Conjecture 19 stated in [É. Sopena, J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327-2333]