78 research outputs found
Almost-rainbow edge-colorings of some small subgraphs
Let be the minimum number of colors necessary to color the edges
of so that every is at least -colored. We improve current bounds
on the {7/4}n-3{5/6}n+1\leq
f(n,4,5)n\not\equiv 1 \pmod 3f(n,4,5)\leq n-1G=K_{n,n}GC_4\subseteq G$ is colored by at least three
colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi,
and D. Mubayi.Comment: 13 page
Lobsters with an almost perfect matching are graceful
Let be a lobster with a matching that covers all but one vertex. We show
that in this case, is graceful.Comment: 4 page
On a Vizing-type integer domination conjecture
Given a simple graph , a dominating set in is a set of vertices
such that every vertex not in has a neighbor in . Denote the domination
number, which is the size of any minimum dominating set of , by .
For any integer , a function
is called a \emph{-dominating function} if the sum of its function
values over any closed neighborhood is at least . The weight of a
-dominating function is the sum of its values over all the vertices. The
-domination number of , , is defined to be the
minimum weight taken over all -domination functions. Bre\v{s}ar,
Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like
problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked
whether there exists an integer so that . In this note we use the Roman -domination number,
of Chellali, Haynes, Hedetniemi, and McRae, (Roman
-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp.
22-28.) to prove that if is a claw-free graph and is an arbitrary
graph, then , which also implies the conjecture for all .Comment: 8 page
All trees are six-cordial
For any integer , a tree is -cordial if there exists a labeling
of the vertices of by , inducing a labeling on the edges with
edge-weights found by summing the labels on vertices incident to a given edge
modulo so that each label appears on at most one more vertex than any other
and each edge-weight appears on at most one more edge than any other.
We prove that all trees are six-cordial by an adjustment of the test proposed
by Hovey (1991) to show all trees are -cordial.Comment: 16 pages, 12 figure
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