Narrow Results

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A-{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b : V(G) ↦ A satisfying Σ_υ∈V(G)b(υ) = 0, there is a function f : E(G) ↦ A* such that for each vertex υ ∈ V(G), the total amount of f values on the edges directed out from υ minus the total amount of f values on the edges directed into υ equals b(υ). Let Z₃ denote the group of order 3. Jaeger et al. conjectured that there exists an integer k such that every k-edge-connected graph is Z₃-connected. In this paper, we prove that every N₂-locally connected claw-free graph G with minimum degree δ(G) ≥ 7 is Z₃-connected.

A total coloring of a graph is an assignment of colors to all the elements of the graph such that no two adjacent or incident elements receive the same color. A graph $G$ is prismatic, if for every triangle $T$, every vertex not in $T$ has exactly one neighbor in $T$. In this paper, we prove the total coloring conjecture (TCC) for prismatic graphs and the tight bound of the TCC for some classes of prismatic graphs.

A total coloring of a graph is an assignment of colors to all the elements (vertices and edges) of the graph such that no two adjacent or incident elements receive the same color. A claw-free graph is a graph that does not have $K_{1,3}$ as an induced subgraph. Quasi-line and inflated graphs are two well-known classes of claw-free graphs. In this paper, we prove that the quasi-line and inflated graphs are totally colorable. In particular, we prove the tight bound of the total chromatic number of some classes of quasi-line graphs and inflated graphs.