MSOL-Definability Equals Recognizability for Halin Graphs and Bounded Degree k-Outerplanar Graphs

One of the most famous algorithmic meta-theorems states that every graph property that can be defined by a sentence in counting monadic second order logic (CMSOL) can be checked in linear time for graphs of bounded treewidth, which is known as Courcelle's Theorem. These algorithms are constructed as finite state tree automata, and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. We prove this conjecture for a number of special cases in a stronger form. That is, we show that each recognizable property is definable in MSOL, i.e. the counting operation is not needed in our expressions. We give proofs for Halin graphs, bounded degree k-outerplanar graphs and some related graph classes. We furthermore show that the conjecture holds for any graph class that admits tree decompositions that can be defined in MSOL, thus providing a useful tool for future proofs

On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if $b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most $6$; in particular we settle this conjecture for cubic Halin graphs.Comment: 18 page

The incidence game chromatic number

We introduce the incidence game chromatic number which unifies the ideas of game chromatic number and incidence coloring number of an undirected graph. For k-degenerate graphs with maximum degree D, the upper bound 2D+4k-2 for the incidence game chromatic number is given. If D is at least 5k, we improve this bound to the value 2D+3k-1. We also determine the exact incidence game chromatic number of cycles, stars and sufficiently large wheels and obtain the lower bound 3D/2 for the incidence game chromatic number of graphs of maximum degree D

Some Results on incidence coloring, star arboricity and domination number

Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.Comment: 8 page

Digraph Coloring Games and Game-Perfectness

In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix