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

F-WORM colorings: Results for 2-connected graphs

Given two graphs F and G, an F-WORM coloring of G is an assignment of colors to its vertices in such a way that no F-subgraph of G is monochromatic or rainbow. If G has at least one such coloring, then it is called F-WORM colorable and W−(G,F) denotes the minimum possible number of colors. Here, we consider F-WORM colorings with a fixed 2-connected graph F and prove the following three main results: (1) For every natural number k, there exists a graph G which is F-WORM colorable and W−(G,F)=k; (2) It is NP-complete to decide whether a graph is F-WORM colorable; (3) For each k≥|V(F)|−1, it is NP-complete to decide whether a graph G satisfies W−(G,F)≤k. This remains valid on the class of F-WORM colorable graphs of bounded maximum degree. We also prove that for each n≥3, there exists a graph G and integers r and s such that s≥r+2, G has Kn-WORM colorings with exactly r and also with s colors, but it admits no Kn-WORM colorings with exactly r+1,…,s−1 colors. Moreover, the difference s−r can be arbitrarily large. © 2017 Elsevier B.V

When the vertex coloring of a graph is an edge coloring of its line graph - a rare coincidence

The 3-consecutive vertex coloring number psi(3c)(G) of a graph G is the maximum number of colors permitted in a coloring of the vertices of G such that the middle vertex of any path P-3 subset of G has the same color as one of the ends of that P-3. This coloring constraint exactly means that no P-3 subgraph of G is properly colored in the classical sense. The 3-consecutive edge coloring number psi(3c)'(G) is the maximum number of colors permitted in a coloring of the edges of G such that the middle edge of any sequence of three edges (in a path P-4 or cycle C-3) has the same color as one of the other two edges. For graphs G of minimum degree at least 2, denoting by L(G) the line graph of G, we prove that there is a bijection between the 3-consecutive vertex colorings of G and the 3-consecutive edge colorings of L(G), which keeps the number of colors unchanged, too. This implies that psi(3c)(G) = psi(3c)'(L(G)); i.e., the situation is just the opposite of what one would expect for first sight

Neighborhood-Restricted Achromatic Colorings of Graphs

A (closed) neighborhood-restricted 2-achromatic-coloring of a graph G is an assignment of colors to the vertices of G such that no more than two colors are assigned in any closed neighborhood. In other words, for every vertex v in G, the vertex v and its neighbors are in at most two different color classes. The 2-achromatic number is defined as the maximum number of colors in any 2-achromatic-coloring of G. We study the 2-achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds

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

Vertex coloring without large polychromatic stars

AbstractGiven an integer k≥2, we consider vertex colorings of graphs in which no k-star subgraph Sk=K1,k is polychromatic. Equivalently, in a star-[k]-coloring the closed neighborhood N[v] of each vertex v can have at most k different colors on its vertices. The maximum number of colors that can be used in a star-[k]-coloring of graph G is denoted by χ̄k⋆(G) and is termed the star-[k] upper chromatic number of G.We establish some lower and upper bounds on χ̄k⋆(G), and prove an analogue of the Nordhaus–Gaddum theorem. Moreover, a constant upper bound (depending only on k) can be given for χ̄k⋆(G), provided that the complement G¯ admits a star-[k]-coloring with more than k colors