Maximal chromatic polynomials of connected planar graphs

Abstract

In this paper we obtain chromatic polynomials of connected 3-and 4-chromatic planar graphs that are maximal for positive integer-valued arguments. We also characterize the class of connected 3-chromatic graphs having the maximum number of pcolorings for p 2 3, thus extending a previous result by the author (the case p = 3).

Maximizing the number of q -colorings

“…We close by mentioning some related work. Tomescu [26][27][28][29][30][31][32][33] and Dohmen [11,12] considered the problem of maximizing or minimizing the number of q-colorings of G given some other parameters, such as chromatic number, connectedness, planarity, and girth. Wright [36] asymptotically determined the total number of q-colored labeled n-vertex graphs with m edges, for the entire range of m; this immediately gives an asymptotic approximation for the average value of P G (q) over all labeled n-vertex graphs with m edges.…”

New bounds for chromatic polynomials and chromatic roots

“…Tomescu settled the problem for x = k = 3 in [6] and later extended it for x ≥ k = 3 in [9] by showing that if G is a graph in C 3 (n) then…”

“…If G ∈ C * k (n) then it is known that (see, for example, [9]) π (G, x) = (x) ↓k (x − 1) n−k as one can first colour the clique of order k and then recursively colour the remaining vertices (which have only one coloured neighbour). On the other hand, …”

Some new bounds for the maximum number of vertex colorings of a (v,e)-graph

“…In [19], he gave a maximum for P (G, λ) in terms of the order of G, where G was a connected graph and λ = 2 or 3. In [21], Tomescu used an ordering of the vertices (an idea used in this paper as well) to help obtain an upper bound for P (G, 4), where G is a connected planar 4-chromatic graph. More recently, in [22] he maximized P (G, λ) for 2-connected graphs of fixed order, and showed that the cycles C n with n vertices are the unique extremal graphs for λ ≥ 3 and n / = 5.…”