152,436 research outputs found
The complement of proper power graphs of finite groups
For a finite group , the proper power graph of is
the graph whose vertices are non-trivial elements of and two vertices
and are adjacent if and only if and or for some
positive integer . In this paper, we consider the complement of
, denoted by . We classify all
finite groups whose complement of proper power graphs is complete, bipartite, a
path, a cycle, a star, claw-free, triangle-free, disconnected, planar,
outer-planar, toroidal, or projective. Among the other results, we also
determine the diameter and girth of the complement of proper power graphs of
finite groups.Comment: 29 pages, 14 figures, Lemma 4.1 has been added and consequent changes
have been mad
A polynomial version of Cereceda's conjecture
Let k and d be such that k ≥ d + 2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d + 2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n 2). So far, the existence of a polynomial diameter is open even for d = 2. In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(n d+1) for k ≥ d + 2. Moreover, we prove that if k ≥ 3 2 (d + 1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k ≥ 2d + 1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs
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
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