Improper coloring of graphs on surfaces

Improper coloring of unit disk graphs

Improper Twin Edge Coloring of Graphs

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

A note on list improper coloring of plane graphs

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v ∈ V (G). An (L, d)-coloring is a mapping φ that assigns a color φ(v) ∈ L(v) to each vertex v ∈ V (G) such that at most d neighbors of v receive color φ(v). A graph is called (k, d)-choosable, if G admits an (L, d)-coloring for every list assignment L with |L(v)| ≥ k for all v ∈ V (G). In th...

On 11-improper 22-coloring of sparse graphs

A graph G is (1, 1)-colorable if its vertices can be partitioned into subsets V1 and V2 so that every vertex in G[Vi] has degree at most 1 for each i ∈ {1, 2}. We prove that every graph with maximum average degree at most 14 5 is (1, 1)-colorable. In particular, it follows that every planar graph with girth at least 7 is (1, 1)-colorable. On the other hand, we construct graphs with maximum aver...

Three-coloring triangle-free graphs on surfaces

Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that X f face of G (|f | − 4) ≤ κ(g + t+ c− 1) for a fixed constant κ, thus generalizing and strengthening several known results. As a corollary, we prove that every triangle-free graph G embedded in a surface of genus g contains a set of...