K3K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161--173] who asked whether every such graph has a $K_3$-WORM coloring with two colors. In fact for every integer $k\ge 3$ there exists a $K_3$-WORM colorable graph in which the minimum number of colors is exactly $k$. There also exist $K_3$-WORM colorable graphs which have a $K_3$-WORM coloring with two colors and also with $k$ colors but no coloring with any of $3,\dots,k-1$ colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide $k$-colorability for every $k \ge 2$ (and remains intractable even for graphs of maximum degree 9 if $k=3$). On the other hand, we prove positive results for $d$-degenerate graphs with small $d$, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.Comment: 18 page

Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

F-WORM colorings of some 2-trees: partition vectors

A collection of distinct subgraphs of a graphG = (V;E).An F-WORM coloring of G is the coloring of its vertices such that no copy of each subgraphFi 2 F is monochrome or rainbow. This generalizes the notion of F-WORMcoloring that was introduced recently by W. Goddard, K. Wash, and H. Xu. A (restricted)partition vector is a sequence whose terms r are the number of F-WORMcolorings using exactly r colors, with. The partition vectors of complete graphsand those of some 2-trees are discussed. We show that, although 2-trees admit the samepartition vector in classic proper vertex colorings which forbid monochrome K2, their partition vectors in K3-WORM colorings are different

Homogeneous colourings of graphs

summary:A proper vertex $k$-colouring of a graph $G$ is called $l$-homogeneous if the number of colours in the neigbourhood of each vertex of $G$ equals $l$. We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs

Homogeneous colourings of graphs

A proper vertex $k$-colouring of a graph $G$ is called $l$-homogeneous if the number of colours in the neigbourhood of each vertex of $G$ equals $l$. We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs

Locating-dominating sets: From graphs to oriented graphs

A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u, v is not an element of S, the neighbourhood of u and v on S are distinct (i.e. N(u) & cap; S &NOTEQUexpressionL; N(v) & cap; S). Locating-dominating sets have received a considerable attention in the last decades. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set S such that for each w is an element of V \ S, N-(w) & cap; S &NOTEQUexpressionL; Phi and for each pair of distinct vertices u, v is an element of V \ S, N-(u) & cap; S &NOTEQUexpressionL; N-(v) & cap; S. We consider the following two parameters. Given an undirected graph G, we look for (gamma)over the arrow(LD) (G) ((gamma)over the arrow(LD) (G)) which is the size of the smallest (largest) optimal locating-dominating set over all orientations of G. In particular, if D is an orientation of G, then (gamma)over the arrow(LD)(G) = gamma(LD)(G) and some for which (gamma)over the arrow(LD)(G) = alpha(G). Finally, we show that for many graph classes (gamma)over the arrow(LD)(G) is polynomial on n but we leave open the question whether there exist graphs with (gamma)over the arrow(LD)(G) is an element of O (log n). (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</p

Center for Space Microelectronics Technology

The 1990 technical report of the Jet Propulsion Laboratory Center for Space Microelectronics Technology summarizes the technical accomplishments, publications, presentations, and patents of the center during 1990. The report lists 130 publications, 226 presentations, and 87 new technology reports and patents