Dong, F. M.

Let G be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the ow polynomial F(G; ) of G are integers if and only if G is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph G for which F(G; ) has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for G is positive if and only if F(G; ) does not have any real root in the interval (1; 2). We also prove that for any non-separable and 3-edge connected G, if G - e is also non-separable for each edge e in G and every 3-edge-cut of G consists of edges incident with some vertex of G, then all roots of P(G; ) are real if and only if either G 2 fL;Z3;K4g or F(G; ) contains at least 9 real roots in the interval (1; 2), where L is the graph with one vertex and one loop and Z3 is the graph with two vertices and three parallel edges joining these two vertices.

For any graph G, let W(G) be the set of vertices in G of degrees larger than 3. We show that for any bridgeless graph G, if W(G) is dominated by some component of G-W(G), then F(G,λ ) has no roots in (1; 2), where F(G,λ ) is the flow polynomial of G. This result generalizes the known result that F(G,λ ) has no roots in (1, 2) whenever |W(G)| ≤2. We also give some constructions to generate graphs whose flow polynomials have no roots in (1, 2).

Let (a1, a2, · · · , ak) denote the graph obtained by connecting two distinct vertices with k independent paths of lengths a1, a2, · · · , ak respectively. Assume that 2 ≤ a1 ≤ a2 ≤ · · · ≤ ak. We prove that the graph θ (a1, a2, · · · , ak) is chromatically unique if ak < a1 +a2, and find examples showing that θ (a1, a2, · · · , ak) may not be chromatically unique if ak = a1 + a2.

For any positive integers a, b, c, d, let Ra,b,c,d be the graph obtained from the complete graphs Ka,Kb,Kc and Kd by adding edges joining every vertex in Ka and Kc to every vertex in Kb and Kd. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of Ra,b,c,d is either a real number or a non-real number with its real part equal to (a + b + c + d − 1)/2.

Chromatic polynomials of graphs have been studied extensively for around one century. The concept of chromatic polynomial of a hypergraph is a natural extension of chromatic polynomial of a graph. It also has been studied for more than 30 years. This short article will focus on introducing some important open problems on chromatic polynomials of hypergraphs.

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

Let 𝐺=(𝑉,𝐸) be a simple graph with 𝑉={1,2,…,𝑛} and 𝜒(𝐺,𝑥) be its chromatic polynomial. For an ordering 𝜋=(𝑣1,𝑣2,…,𝑣𝑛) of elements of 𝑉 , let 𝛿𝐺(𝜋) be the number of integers 𝑖 , where 1≤𝑖≤𝑛−1 , with either 𝑣𝑖<𝑣𝑖+1 or 𝑣𝑖𝑣𝑖+1∈𝐸 . Let 𝒲(𝐺) be the set of subsets {𝑎,𝑏,𝑐} of 𝑉 , where 𝑎<𝑏<𝑐 , which induces a subgraph of 𝐺 with 𝑎𝑐 as its only edge. We show that 𝒲(𝐺) =∅ if and only if (−1)𝑛𝜒(𝐺,−𝑥)=∑𝜋(𝑥+𝛿𝐺(𝜋)𝑛) , where the sum runs over all 𝑛! orderings 𝜋 of 𝑉 . To prove this result, we establish an analogous result on order polynomials of posets and apply Stanley's work on the relation between chromatic polynomials and order polynomials.