A total coloring of a graph G is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An interval total t-coloring of a graph G is a total coloring of G with colors 1, 2,. .. , t such that at least one vertex or edge of G is colored by i, i = 1, 2,. .. , t, and the edges incident to each vertex v together with v ar...

A proper edge coloring of a simple graph G is called vertex-distinguishing if no two distinct vertices are incident to the same set of colors. We prove that the minimum number of colors required for a vertex-distinguishing coloring of a random graph of order n is almost always equal to the maximum degree ∆(G) of the graph.

A signed tree-coloring of a signed graph (G, σ) is a vertex coloring c so that G(i,±) is a forest for every i ∈ c(u) and u ∈ V(G), where G(i,±) is the subgraph of (G, σ) whose vertex set is the set of vertices colored by i or −i and edge set is the set of positive edges with two end-vertices colored both by i or both by −i, along with the set of negative edges with one end-vertex colored by i a...

An equitable two-coloring of a hypergraph H = (V, E) is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with property B problem Radhakrishnan and Srinivasan proved that if H is a k-uniform hypergraph with maximum vertex degree ∆(H) satisfying ∆(H) c 2 k−1 √ k ln k (1) for some absolute constant c > 0, then H is 2-colorable. By using...

An adjacent vertex distinguishing edge-coloring or an avd-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree ∆ and with no isolated edges has an avd-coloring with at most ∆ + 300 colors, provided that ∆ > 1020. AMS Subject Classification: 05C15

An ℓ-facial edge-coloring of a plane graph is coloring its edges such that any two at distance most ℓ on boundary walk face receive distinct colors. It the variant vertex coloring, which arose as generalization well-known cyclic coloring. conjectured 3ℓ+1 colors suffice for an graph. The conjecture has only been confirmed ℓ≤2, and in this paper, we prove validity ℓ=3.

A b-coloring of a graph is proper coloring its vertices such that each color class contains vertex adjacent to at least one every other class. The b-chromatic number the largest integer k has with colors. In this work we present lower bounds for vertex-deleted subgraph graph, particularly regarding two important classes, claw-free graphs and chordal graphs. We also get G−{x}, when G large girth.