The Edge-Distinguishing Chromatic Number of Petal Graphs, Chorded Cycles, and Spider Graphs

The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges $uv$. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.Comment: 23 pages, 1 figur

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