Harmonious Coloring of Trees with Large Maximum Degree

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a harmonious coloring of $G$. We show that if $T$ is a forest of order $n$ with maximum degree $\Delta(T)\geq \frac{n+2}{3}$, then h(T)= \Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$; \Delta(T)+1, & otherwise. Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.Comment: 8 pages, 1 figur

Extremal problems involving forbidden subgraphs

In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs. In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free $(0,k)$-colorable graphs and we find the asymptotic threshold of density of $(j,k)$-colorable graphs of large girth when $k\geq 2j+2$. In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of $r$-dynamic chromatic number of graphs in terms of other parameters. In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs. In particular, we characterize hypergraphs with the maximum number of edges that contain no $r$-regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs

Asymmetric coloring games on incomparability graphs

Consider the following game on a graph $G$: Alice and Bob take turns coloring the vertices of $G$ properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of $G$ is the minimum number of colors that allows Alice to win the game. The game Grundy number of $G$ is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The $(a,b)$-game chromatic and Grundy numbers are defined likewise except that Alice colors $a$ vertices and Bob colors $b$ vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs $(a,b)$ for which the $(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu