Distinguishing colorings of graphs and their subgraphs

In this paper, several distinguishing colorings of graphs are studied, such as vertex distinguishing proper edge coloring, adjacent vertex distinguishing proper edge coloring, vertex distinguishing proper total coloring, adjacent vertex distinguishing proper total coloring. Finally, some related chromatic numbers are determined, especially the comparison of the correlation chromatic numbers between the original graph and the subgraphs are obtained

Algebraic Analysis of Vertex-Distinguishing Edge-Colorings

Vertex-distinguishing edge-colorings (vdec colorings) are a restriction of proper edge-colorings. These special colorings require that the sets of edge colors incident to every vertex be distinct. This is a relatively new field of study. We present a survey of known results concerning vdec colorings. We also define a new matrix which may be used to study vdec colorings, and examine its properties. We find several bounds on the eigenvalues of this matrix, as well as results concerning its determinant, and other properties. We finish by examining related topics and open problems

Graph Colorings with Constraints

A graph is a collection of vertices and edges, often represented by points and connecting lines in the plane. A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Here we examine results from three different coloring problems. First, adjacent vertex distinguishing total colorings are proper total colorings such that the set of colors appearing at each vertex is distinct for every pair of adjacent vertices. Next, vertex coloring total weightings are an assignment of weights to the vertices and edges of a graph so that every pair of adjacent vertices have distinct weight sums. Finally, edge list multi-colorings consider assignments of color lists and demands to edges; edges are colored with a subset of their color list of size equal to its color demand so that adjacent edges have disjoint sets. Here, color sets consisting of measurable sets are considered

Color-blind index in graphs of very low degree

Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in $G$ in nonincreasing order to obtain $c^*(v)$, the color-blind partition of $v$. When $c^*$ induces a proper vertex coloring, that is, $c^*(u)\neq c^*(v)$ for every edge $uv$ in $G$, we say that $c$ is color-blind distinguishing. The minimum $k$ for which there exists a color-blind distinguishing edge coloring $c:E(G)\to [k]$ is the color-blind index of $G$, denoted $\operatorname{dal}(G)$. We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if $\operatorname{dal}(G) \leq 2$ is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when $\operatorname{dal}(G)$ is finite for a class of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi

List Distinguishing Parameters of Trees

A coloring of the vertices of a graph G is said to be distinguishing} provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, chi_D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of lists of available colors to the vertices of G, we say that G is (properly) L-distinguishable if there is a (proper) distinguishing coloring f of G such that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is the minimum integer k such that G is L-distinguishable for any list assignment L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is properly L-distinguishable for any list assignment L with |L(v)| = k for all v. In this paper, we study these distinguishing parameters for trees, and in particular extend an enumerative technique of Cheng to show that for any tree T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.Comment: 10 page