On Harmonious Colouring of Trees

Abstract

Let $G$ be a simple graph and $\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\Delta(T)\geq\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\Delta(T) \le \Big\lceil\frac{n}{2}\Big\rceil$, then there exists a harmonious colouring of $T$ with $\Big\lceil \frac{n}{2}\Big \rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\leq \Big\lceil \frac{n}{2} \Big\rceil +1$.

Harmonious coloring of trees with large maximum degree

“…Suppose v has exactly p adjacent non-leaves in T . We claim that a contradiction to (1). By (8), we can place all the vertices of P and all remaining non-leaf neighbors of v into distinct vertices of H. After that, we place the leaves adjacent to v into distinct vertices of H not containing v or its neighbors.…”

“…Since vertices at distance at most two in a graph G must have distinct colors in any harmonious coloring of G, h(G) ≥ ∆(G) + 1 for every graph G. In [1] it was shown that if T is a tree of order n and ∆(T ) ≥ n 2 , then h(T ) = ∆(T ) + 1. Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a tree.…”

The Edge-Distinguishing Chromatic Number of Petal Graphs, Chorded Cycles, and Spider Graphs

“…We are going to define the embedding φ : V (S 1 , 2 , 3 , 4 ) → V (K * 4 ) explicitly. When k = 4, since n = 10 and 1 = 2, ( 1 , 2 , 3 , 4 ) ∈ {(1, 1, 1, 7), (1, 1, 2, 6), (1,1,3,5), (1,1,4,4), (1, 2, 2, 5), (1,2,3,4), (1,3,3,3)}.…”

The edge-distinguishing chromatic number of petal graphs, chorded cycles, and spider graphs

“…Determining λ(G) for a general graph G is NP-complete [13]. Consequently, most work in the literature focuses on providing bounds on h(G) and thus λ(G) [1] or studying the asymptotic behavior of h(G) [6] for various families of graphs G. When trying to determine the exact formula for λ(G) or h(G), only very limited families have been tackled. For example, λ(G) is determined for paths and cycles [3,12], and h(G) is determined for complete r-ary trees [10].…”