5 research outputs found
Dynamic Chromatic Number of Regular Graphs
A dynamic coloring of a graph is a proper coloring such that for every
vertex of degree at least 2, the neighbors of receive at least
2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}.
PhD thesis, West Virginia University, 2001.] that if is a -regular
graph, then . In this paper, we prove that if is a
-regular graph with , then . It confirms the conjecture for all regular graph with
diameter at most 2 and . In fact, it shows that
provided that has diameter at most 2 and
. Moreover, we show that for any -regular graph ,
. Also, we show that for any there exists a
regular graph whose chromatic number is and .
This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A.
Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number
and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In
press].Comment: 8 page
Colorings of complements of line graphs
Our purpose is to show that complements of line graphs enjoy nice coloring
properties. We show that for all graphs in this class the local and usual
chromatic numbers are equal. We also prove a sufficient condition for the
chromatic number to be equal to a natural upper bound. A consequence of this
latter condition is a complete characterization of all induced subgraphs of the
Kneser graph that have a chromatic number equal to its
chromatic number, namely . In addition to the upper bound, a lower bound
is provided by Dol'nikov's theorem, a classical result of the topological
method in graph theory. We prove the -hardness of deciding
the equality between the chromatic number and any of these bounds.
The topological method is especially suitable for the study of coloring
properties of complements of line graphs of hypergraphs. Nevertheless, all
proofs in this paper are elementary and we also provide a short discussion on
the ability for the topological methods to cover some of our results
Rainbow paths with prescribed ends
It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if G is a connected graph distinct from C7, then there is a χ(G)-coloring of G in which every vertex v ∈ V (G) is an initial vertex of a path P with χ(G) vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs. Electron. J. Combin. 18(1):P17, 9pp, 2011] this was proved with ⌊ χ(G) 2 ⌋ vertices instead of χ(G) vertices. We strengthen this to χ(G) − 1 vertices. We also prove that every connected graph with at least one edge has a proper k-coloring (for some k) such that every vertex of color i has a neighbor of color i + 1 (mod k). C5 shows that k may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the k-coloring exists for every k ≥ χ(G). In fact, the k-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by 1 (mod k) along each edge. The method is based on the circular chromatic number χc(G). In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number