10 research outputs found
Facial unique-maximum colorings of plane graphs with restriction on big vertices
A facial unique-maximum coloring of a plane graph is a proper coloring of the
vertices using positive integers such that each face has a unique vertex that
receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed
a strengthening of the Four Color Theorem conjecturing that all plane graphs
have a facial unique-maximum coloring using four colors. This conjecture has
been disproven for general plane graphs and it was shown that five colors
suffice. In this paper we show that plane graphs, where vertices of degree at
least four induce a star forest, are facially unique-maximum 4-colorable. This
improves a previous result for subcubic plane graphs by Andova, Lidick\'y,
Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some
problems.Comment: 8 pages, 5 figure
On facial unique-maximum (edge-)coloring
A facial unique-maximum coloring of a plane graph is a vertex coloring where
on each face the maximal color appears exactly once on the vertices of
. If the coloring is required to be proper, then the upper bound for
the minimal number of colors required for such a coloring is set to .
Fabrici and G\"oring [Fabrici and Goring 2016] even conjectured that colors
always suffice. Confirming the conjecture would hence give a considerable
strengthening of the Four Color Theorem. In this paper, we prove that the
conjecture holds for subcubic plane graphs, outerplane graphs and plane
quadrangulations. Additionally, we consider the facial edge-coloring analogue
of the aforementioned coloring and prove that every -connected plane graph
admits such a coloring with at most colors.Comment: 5 figure
A counterexample to a conjecture on facial unique-maximal colorings
A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face α the maximal color appears exactly once on the vertices of α. Fabrici and Göring [4] proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color theorem. Wendland [6] later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. s we conclude that facial unique-maximum chromatic number of the sphere is five
Coloring problems in graph theory
In this thesis, we focus on variants of the coloring problem on graphs. A coloring of a graph is an assignment of colors to the vertices. A coloring is proper if no two adjacent vertices are assigned the same color. Colorings are a central part of graph theory and over time many variants of proper colorings have been introduced. The variants we study are packing colorings, improper colorings, and facial unique-maximum colorings.
A packing coloring of a graph is an assignment of colors to the vertices of such that the distance between any two vertices that receive color is greater than . A -coloring of is an assignment of colors to the vertices of such that the distance between any two vertices that receive color is greater than . We study packing colorings of multi-layer hexagonal lattices, improving a result of Fiala, Klav\v{z}ar, and Lidick\\u27{y}, and find the packing chromatic number of the truncated square lattice. We also prove that subcubic planar graphs are -colorable.
A facial unique-maximum coloring of is an assignment of colors to the vertices of such that no two adjacent vertices receive the same color and the maximum color on a face appears only once on that face. We disprove a conjecture of Fabrici and G\ {o}ring that plane graphs are facial unique-maximum -colorable. Inspired by this result, we also provide sufficient conditions for the facial unique-maximum -colorability of a plane graph.
A -coloring of is an assignment of colors and to the vertices of such that the vertices that receive color form an independent set and the vertices that receive color form a linear forest. We will explore -colorings, an offshoot of improper colorings, and prove that subcubic planar -free graphs are -colorable. This result is a corollary of a theorem by Borodin, Kostochka, and Toft, a fact that we failed to realize before the completion of our proof
Proper connection number of graphs
The concept of \emph{proper connection number} of graphs is an extension of proper colouring and is motivated by rainbow connection number of graphs. Let be an edge-coloured graph. Andrews et al.\cite{Andrews2016} and, independently, Borozan et al.\cite{Borozan2012} introduced the concept of proper connection number as follows: A coloured path in an edge-coloured graph is called a \emph{properly coloured path} or more simple \emph{proper path} if two any consecutive edges receive different colours. An edge-coloured graph is called a \emph{properly connected graph} if every pair of vertices is connected by a proper path. The \emph{proper connection number}, denoted by , of a connected graph is the smallest number of colours that are needed in order to make properly connected. Let be an integer. If every two vertices of an edge-coloured graph are connected by at least proper paths, then is said to be a \emph{properly -connected graph}. The \emph{proper -connection number} , introduced by Borozan et al. \cite{Borozan2012}, is the smallest number of colours that are needed in order to make a properly -connected graph.
The aims of this dissertation are to study the proper connection number and the proper 2-connection number of several classes of connected graphs. All the main results are contained in Chapter 4, Chapter 5 and Chapter 6.
Since every 2-connected graph has proper connection number at most 3 by Borozan et al. \cite{Borozan2012} and the proper connection number of a connected graph equals 1 if and only if is a complete graph by the authors in \cite{Andrews2016, Borozan2012}, our motivation is to characterize 2-connected graphs which have proper connection number 2. First of all, we disprove Conjecture 3 in \cite{Borozan2012} by constructing classes of 2-connected graphs with minimum degree that have proper connection number 3. Furthermore, we study sufficient conditions in terms of the ratio between the minimum degree and the order of a 2-connected graph implying that has proper connection number 2. These results are presented in Chapter 4 of the dissertation.
In Chapter 5, we study proper connection number at most 2 of connected graphs in the terms of connectivity and forbidden induced subgraphs , where are three integers and (where is the graph consisting of three paths with and edges having an end-vertex in common).
Recently, there are not so many results on the proper -connection number , where is an integer. Hence, in Chapter 6, we consider the proper 2-connection number of several classes of connected graphs. We prove a new upper bound for and determine several classes of connected graphs satisfying . Among these are all graphs satisfying the Chv\'{a}tal and Erd\'{o}s condition ( with two exceptions). We also study the relationship between proper 2-connection number and proper connection number of the Cartesian product of two nontrivial connected graphs.
In the last chapter of the dissertation, we propose some open problems of the proper connection number and the proper 2-connection number
Unique-Maximum Coloring Of Plane Graphs
A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . . , k so that, for each face of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors
Unique-Maximum Coloring Of Plane Graphs
A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . ., k so that, for each face of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors