On Locally Identifying Coloring of Cartesian Product and Tensor Product of Graphs

For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping $f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each edge $uv \in E(G)$. The smallest integer $k$ for which there is a proper $k$-coloring of $G$ is called chromatic number of $G$, denoted by $\chi(G)$. A \emph{locally identifying coloring} (for short, lid-coloring) of a graph $G$ is a proper $k$-coloring of $G$ such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer $k$ such that $G$ has a lid-coloring with $k$ colors is called \emph{locally identifying chromatic number} (for short, \emph{lid-chromatic number}) of $G$, denoted by $\chi_{lid}(G)$. In this paper, we study lid-coloring of Cartesian product and tensor product of two graphs. We prove that if $G$ and $H$ are two connected graphs having at least two vertices then (a) $\chi_{lid}(G \square H) \leq \chi(G) \chi(H)-1$ and (b) $\chi_{lid}(G \times H) \leq \chi(G) \chi(H)$. Here $G \square H$ and $G \times H$ denote the Cartesian and tensor products of $G$ and $H$ respectively. We also give exact values of lid-chromatic number of Cartesian product (resp. tensor product) of two paths, a cycle and a path, and two cycles

Coloring Grids Avoiding Bicolored Paths

The vertex-coloring problem on graphs avoiding bicolored members of a family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of graphs (Gr\"unbaum, 1973) where bicolored copies of $P_4$ and cycles are not allowed, respectively. In this paper, we study a variation of this problem, by considering vertex coloring on grids forbidding bicolored paths. We let $P_k$-chromatic number of a graph be the minimum number of colors needed to color the vertex set properly avoiding a bicolored $P_k.$ We show that in any 3-coloring of the cartesian product of paths, $P_{k-2}\square P_{k-2}$, there is a bicolored $P_k.$ With our result, the problem of finding the $P_k$-chromatic number of product of two paths (2-dimensional grid) is settled for all $k.

Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure

Cartesian product of hypergraphs: properties and algorithms

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspects of Cartesian products of hypergraphs. We also extend a classical prime factorization algorithm initially designed for graphs to connected conformal hypergraphs using 2-sections of hypergraphs