2,521 research outputs found
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
Interval total colorings of graphs
A total coloring of a graph is a coloring of its vertices and edges such
that no adjacent vertices, edges, and no incident vertices and edges obtain the
same color. An \emph{interval total -coloring} of a graph is a total
coloring of with colors such that at least one vertex or edge
of is colored by , , and the edges incident to each vertex
together with are colored by consecutive colors, where
is the degree of the vertex in . In this paper we investigate
some properties of interval total colorings. We also determine exact values of
the least and the greatest possible number of colors in such colorings for some
classes of graphs.Comment: 23 pages, 1 figur
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