49 research outputs found
Solution of Vizing's Problem on Interchanges for Graphs with Maximum Degree 4 and Related Results
Let be a Class 1 graph with maximum degree and let be an
integer. We show that any proper -edge coloring of can be transformed to
any proper -edge coloring of using only transformations on -colored
subgraphs (so-called interchanges). This settles the smallest previously
unsolved case of a well-known problem of Vizing on interchanges, posed in 1965.
Using our result we give an affirmative answer to a question of Mohar for two
classes of graphs: we show that all proper -edge colorings of a Class 1
graph with maximum degree 4 are Kempe equivalent, that is, can be transformed
to each other by interchanges, and that all proper 7-edge colorings of a Class
2 graph with maximum degree 5 are Kempe equivalent
Latin cubes of even order with forbidden entries
We consider the problem of constructing Latin cubes subject to the condition
that some symbols may not appear in certain cells. We prove that there is a
constant such that if and is a -dimensional array where every cell contains at most symbols, and
every symbol occurs at most times in every line of , then is
{\em avoidable}; that is, there is a Latin cube of order such that for
every , the symbol in position of does not
appear in the corresponding cell of .Comment: arXiv admin note: substantial text overlap with arXiv:1809.0239
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
Improper interval edge colorings of graphs
A -improper edge coloring of a graph is a mapping
such that at most edges of with
a common endpoint have the same color. An improper edge coloring of a graph
is called an improper interval edge coloring if the colors of the edges
incident to each vertex of form an integral interval. In this paper we
introduce and investigate a new notion, the interval coloring impropriety (or
just impropriety) of a graph defined as the smallest such that has
a -improper interval edge coloring; we denote the smallest such by
. We prove upper bounds on for
general graphs and for particular families such as bipartite, complete
multipartite and outerplanar graphs; we also determine
exactly for belonging to some particular classes of graphs. Furthermore, we
provide several families of graphs with large impropriety; in particular, we
prove that for each positive integer , there exists a graph with
. Finally, for graphs with at least two vertices we
prove a new upper bound on the number of colors used in an improper interval
edge coloring
On star edge colorings of bipartite and subcubic graphs
A star edge coloring of a graph is a proper edge coloring with no -colored
path or cycle of length four. The star chromatic index of
is the minimum number for which has a star edge coloring with
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 . We also consider bipartite graphs where all
vertices in one part have maximum degree and all vertices in the other part
has maximum degree . Let be an integer (), we prove that if
then ; and if , then ; both upper bounds are sharp.
Finally, we consider the well-known conjecture that subcubic graphs have star
chromatic index at most ; in particular we settle this conjecture for cubic
Halin graphs.Comment: 18 page
Extending partial edge colorings of iterated cartesian products of cycles and paths
We consider the problem of extending partial edge colorings of iterated
cartesian products of even cycles and paths, focusing on the case when the
precolored edges satisfy either an Evans-type condition or is a matching. In
particular, we prove that if is the th power of the cartesian
product of the even cycle with itself, and at most edges of
are precolored, then there is a proper -edge coloring of that agrees
with the partial coloring. We show that the same conclusion holds, without
restrictions on the number of precolored edges, if any two precolored edges are
at distance at least from each other. For odd cycles of length at least
, we prove that if is the th power of the cartesian
product of the odd cycle with itself (), and at most
edges of are precolored, then there is a proper -edge coloring of
that agrees with the partial coloring. Our results generalize previous ones
on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020)
410--444]