On the degree-chromatic polynomial of a tree

Journal of Combinatorial Theory, Series B, 2000

Let P(G, *) denote the chromatic polynomial of a graph G. It is proved in this paper that for every connected graph G of order n and real number * n, (*&2) n&1 P(G, *)&*(*&1) n&2 P(G, *&1) 0. By this result, the following conjecture proposed by Bartels and Welsh is proved: P(G, n)(P(G, n&1)) &1 >e for every graph G of order n.

Discrete Mathematics, 2000

The tree graph T (G) of a connected graph G has as vertices the spanning trees of G, and two trees are adjacent if one is obtained from the other by interchanging one edge. In this paper we study the chromatic number of T (G) and of a related graph T * (G).

… abstract for 18th Annual Conference on …, 2006

Stanley defined the chromatic symmetric function X(G) of a graph G as a sum of monomial symmetric functions corresponding to proper colorings of G, and asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's question by asking what invariants of a tree T can be recovered from its chromatic symmetric function X(T ). We prove that the degree sequence (δ 1 , . . . ), where δ j is the number of vertices of T of degree j, and the path sequence (π 1 , . . . ), where π k is the number of k-edge paths in T , are given by explicit linear combinations of the coefficients of X(T ). These results are consistent with an affirmative answer to Stanley's question. We briefly present some applications of these results to classifying certain special classes of trees by their chromatic symmetric functions.

Mathematical Programming, 2002

Let P(G, λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ ≥ n,

AKCE International Journal of Graphs and Combinatorics, 2017

For any positive integer k, a k-distance coloring of a graph G is a vertex coloring of G in which no two vertices at distance less than or equal to k receive the same color. The k-distance chromatic number of G, denoted by χ k (G) is the smallest integer α for which G has a k-distance α-coloring. In this paper, we improve the lower bound for the k-distance chromatic number of an arbitrary graph for k odd case and see that trees achieve this lower bound by determining the k-distance chromatic number of trees. Also, we find k-distance chromatic number of cycles and 2-distance chromatic number of a graph G in which every pair of cycles are edge disjoint. c

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in A colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed A [1,4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another. If F n is a full (also known as 'complete') graph on n vertices, & < n , then P(F k c F H , pi, A) = M (*>(A-*)<""*>, where M <*> = M (M-l)(/i-2). .. (/ *-* + 1). In particular P(F t <= F,,, A, A) = A W (A-*)<"-*> = A (n). If £ " is empty (also known as 'null' or 'totally disconnected' graph on n vertices) then P(E k a E, n fi, A) = fi k \"-k , in particular P(E k c £ " , A, A) = A". Let G = (X,V), \X\ = n be a graph and G o = (* " , K o), X 0 ^X 0 \X 0 \ = m be an induced subgraph of G. Let also x,y <= Xbe two non-adjacent vertices in G. We construct the graph G, from G by joining x and _y by an edge and the graph G 2 , obtained from G by contracting x and y into single vertex. Then we can observe that the following equality is true: P(G 0 ^G, ti, A) = P(G l 0 ^G u fi,\) + P(Gl<=G 2 , /i,A), (1) where Gi = Go = G o if x,_y ^ A'o, and G o , Go are the subgraphs induced respectively by X ih X 0 U{x,y} otherwise. It is so because P (G 0 c G 2 , M , A) equals the number of colourings for which x and y have the same colour. If we perform this operation further as much as possible we obtain P(G 0 = G, n, A) = X P{F kl c Fn,, M, A). /=i Since P(F k <= F n ,fj.,A) = P(F' k <^F n ,p,A) for any two fc-vertex complete subgraphs F A ., F' k of F,,, we can write P(G 0 cz G, n, A) = 5 t o, y P(F y ^ F h M , A). In order to clear up the meaning of coefficients a, y we consider one more class of colourings. A colouring of a graph is called an /-colouring, if exactly / colours are used, but Glasgow Math. J. 36 (1994) 265-267.

Discrete Mathematics, 2015

If G is a k-chromatic graph of order n then it is known that the chromatic polynomial of G, π(G, x), is at most x(x − 1) • • • (x − (k − 1))x n−k = (x) ↓k x n−k for every x ∈ N. We improve here this bound by showing that π(G, x) ≤ (x) ↓k (x − 1) ∆(G)−k+1 x n−1−∆(G) for every x ∈ N, where ∆(G) is the maximum degree of G. Secondly, we show that if G is a connected k-chromatic graph of order n where k ≥ 4 then π(G, x) is at most (x) ↓k (x − 1) n−k for every real x ≥ n − 2 + n 2 − k 2 − n + k 2 (it had been previously conjectured that this inequality holds for all x ≥ k). Finally, we provide an upper bound on the moduli of the chromatic roots that is an improvment over known bounds for dense graphs.

2017

Let $G$ be a graph of order $n$. It is well-known that $\alpha(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $\alpha(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ \alpha(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $\alpha(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $\chi(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(\Delta^+(D),\Delta^-(D))$. We give a short and simple proof for this result. Next, we investigate the chromatic number of tournaments and determine the unique tournament such that for every integer $k&gt;1$, the number of proper $k$-colorings of that tournament is maximum among all strongly connected tournaments with the same number of vertices. Also, we find the chromatic polynomial of the strongly connected tournament wit...

Let c be a vertex k-coloring on a connected graph G(V, E). Let Π = {C1, C2, ..., C k } be the partition of V (G) induced by the coloring c. The color code cΠ(v) of a vertex v in G is (d(v, C1), d(v, C2), ..., d(v, C k)), where d(v, Ci) = min{d(v, x)|x ∈ Ci} for 1 ≤ i ≤ k. If any two distinct vertices u, v in G satisfy that cΠ(u) = cΠ(v), then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χL(G), is the smallest k such that G admits a locating k-coloring. Let T (n, k) be a complete n-ary tree, namely a rooted tree with depth k in which each vertex has n children except for its leaves. In this paper, we study the locating-chromatic number of T (n, k) .

Mathematics and Statistics, 2023

In this paper, graphs called overtrees are introduced and studied. These are connected graphs that contain a single simple cycle. Such graphs are connected graphs following the trees in terms of the number of edges. An overtree can be obtained from a tree by adding an edge to connect two non-adjacent vertices of a tree. The same class of graphs can also be defined as a class of graphs obtained from trees by replacing one vertex of the tree with a simple cycle. The main characteristics of overtrees are n, which is the number of vertices, and k, which is the number of vertices of a simple cycle (3 ≤ k ≤ n). A formula for the chromatic polynomial of an overtree is obtained, which is determined by the characteristics n and k only. As a consequence, it is obtained the formula for the chromatic function of a graph which is built from a tree by replacing some of its vertices (possibly all) with simple cycles of arbitrary length. It follows from these formulas that any overtree with an even-length cycle is two-colored, and with an odd-length cycle is three-colored. The same is true for graphs obtained from trees by replacing some vertices with simple cycles.