Distance constraints in graph color extensions

Graph Color Extensions :

Different-Distance Sets in a Graph

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

Graph Color Extensions: When Hadwiger's Conjecture and Embeddings Help

Suppose G is r-colorable and P ⊆ V (G) is such that the components of G[P ] are far apart. We show that any (r + s)-coloring of G[P ] in which each component is s-colored extends to an (r + s)-coloring of G. If G does not contract to K5 or is planar and s ≥ 2, then any (r + s − 1)-coloring of P in which each component is s-colored extends to an (r + s − 1)-coloring of G. This result uses the Fo...

Distance-Based Topological Indices and Double graph

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

Subgraphs Graph in a Distance-regular Graph

Let Γ denote a D-bounded distance-regular graph, where D ≥ 3 is the diameter of Γ. For 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s, define a graph G(∆) whose vertex set is the collection of all weak-geodetically closed subgraphs of diameter s+2 containing ∆, and vertex Ω is adjacent to vertex Ω′ in G if and only if Ω∩Ω′ as a subgraph of Γ has diameter s+1. We sh...