57 research outputs found
Edge-colorings of 4-regular graphs with the minimum number of palettes
A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct
colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Hor\u148\ue1k, Kalinowski, Meszka and Wo\u17aniak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs
Some results on the palette index of graphs
Given a proper edge coloring of a graph , we define the palette
of a vertex as the set of all colors appearing
on edges incident with . The palette index of is the
minimum number of distinct palettes occurring in a proper edge coloring of .
In this paper we give various upper and lower bounds on the palette index of
in terms of the vertex degrees of , particularly for the case when
is a bipartite graph with small vertex degrees. Some of our results concern
-biregular graphs; that is, bipartite graphs where all vertices in one
part have degree and all vertices in the other part have degree . We
conjecture that if is -biregular, then , and we prove that this conjecture holds for several families of
-biregular graphs. Additionally, we characterize the graphs whose
palette index equals the number of vertices
Об индексе палитры треугольника Серпинского и графа Серпинского
The palette of a vertex v of a graph G in a proper edge coloring is the set of colors assigned to the edges which are incident to v. The palette index of G is the minimum number of palettes occurring among all proper edge colorings of G. In this paper, we consider the palette index of Sierpinski graphs S” and Sierpinski triangle graphs S” . In particular, we determine the exact value of the palette index of Sierpinski triangle graphs. We also determine the palette index of Sierpinski graphs S” where p is even, p = 3, or n = 2 and p = 41 + 3
Graphs with large palette index
Given an edge-coloring of a graph, the palette of a vertex is defined as the
set of colors of the edges which are incident with it. We define the palette
index of a graph as the minimum number of distinct palettes, taken over all
edge-colorings, occurring among the vertices of the graph. Several results
about the palette index of some specific classes of graphs are known. In this
paper we propose a different approach that leads to new and more general
results on the palette index. Our main theorem gives a sufficient condition for
a graph to have palette index larger than its minimum degree. In the second
part of the paper, by using such a result, we answer to two open problems on
this topic. First, for every odd, we construct a family of -regular
graphs with palette index reaching the maximum admissible value. After that, we
construct the first known family of simple graphs whose palette index grows
quadratically with respect to their maximum degree.Comment: 7 pages, 1 figur
A Characterization of Graphs with Small Palette Index
Given an edge-coloring of a graph G, we associate to every vertex v of G the set of colors appearing on the edges incident with v. The palette index of G is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of G. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index 1 are r-regular graphs admitting an r-edge-coloring, while regular graphs with palette index 2 do not exist. Here, we characterize all graphs with palette index either 2 or 3 in terms of the existence of suitable decompositions in regular subgraphs. As a corollary, we obtain a complete characterization of regular graphs with palette index 3
A note on one-sided interval edge colorings of bipartite graphs
For a bipartite graph with parts and , an -interval coloring is
a proper edge coloring of by integers such that the colors on the edges
incident to any vertex in form an interval. Denote by
the minimum such that has an -interval coloring with colors. The
author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that
there is a polynomial such that if has maximum degree at most
, then . In this short note, we prove
this conjecture; in fact, we prove that a cubic polynomial suffices. We also
deduce some improved upper bounds on for bipartite graphs
with small maximum degree
Color-blind index in graphs of very low degree
Let be an edge-coloring of a graph , not necessarily
proper. For each vertex , let , where is
the number of edges incident to with color . Reorder for
every in in nonincreasing order to obtain , the color-blind
partition of . When induces a proper vertex coloring, that is,
for every edge in , we say that is color-blind
distinguishing. The minimum for which there exists a color-blind
distinguishing edge coloring is the color-blind index of ,
denoted . We demonstrate that determining the
color-blind index is more subtle than previously thought. In particular,
determining if is NP-complete. We also connect
the color-blind index of a regular bipartite graph to 2-colorable regular
hypergraphs and characterize when is finite for a class
of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi
Locality of not-so-weak coloring
Many graph problems are locally checkable: a solution is globally feasible if
it looks valid in all constant-radius neighborhoods. This idea is formalized in
the concept of locally checkable labelings (LCLs), introduced by Naor and
Stockmeyer (1995). Recently, Chang et al. (2016) showed that in bounded-degree
graphs, every LCL problem belongs to one of the following classes:
- "Easy": solvable in rounds with both deterministic and
randomized distributed algorithms.
- "Hard": requires at least rounds with deterministic and
rounds with randomized distributed algorithms.
Hence for any parameterized LCL problem, when we move from local problems
towards global problems, there is some point at which complexity suddenly jumps
from easy to hard. For example, for vertex coloring in -regular graphs it is
now known that this jump is at precisely colors: coloring with colors
is easy, while coloring with colors is hard.
However, it is currently poorly understood where this jump takes place when
one looks at defective colorings. To study this question, we define -partial
-coloring as follows: nodes are labeled with numbers between and ,
and every node is incident to at least properly colored edges.
It is known that -partial -coloring (a.k.a. weak -coloring) is easy
for any . As our main result, we show that -partial -coloring
becomes hard as soon as , no matter how large a we have.
We also show that this is fundamentally different from -partial
-coloring: no matter which we choose, the problem is always hard
for but it becomes easy when . The same was known previously
for partial -coloring with , but the case of was open
- …