38,353 research outputs found
A new variant of the Erd\H{o}s-Gy\'{a}rf\'{a}s problem on
Motivated by an extremal problem on graph-codes that links coding theory and
graph theory, Alon recently proposed a question aiming to find the smallest
number such that there is an edge coloring of by colors with no
copy of given graph in which every color appears an even number of times.
When , the question of whether colors are enough, was
initially emphasized by Alon. Through modifications to the coloring functions
originally designed by Mubayi, and Conlon, Fox, Lee and Sudakov, the question
of has already been addressed. Expanding on this line of inquiry, we
further study this new variant of the generalized Ramsey problem and provide a
conclusively affirmative answer to Alon's question concerning .Comment: Note added: Heath and Zerbib also proved the result on
independently. arXiv:2307.0131
On Ramsey numbers of complete graphs with dropped stars
Let be the smallest integer such that for any -coloring (say,
red and blue) of the edges of , , there is either a red
copy of or a blue copy of . Let be the complete graph on
vertices from which the edges of are dropped. In this note we
present exact values for and new upper bounds
for in numerous cases. We also present some results for
the Ramsey number of Wheels versus .Comment: 9 pages ; 1 table in Discrete Applied Mathematics, Elsevier, 201
Coloring Sums of Extensions of Certain Graphs
Recall that the minimum number of colors that allow a proper coloring of
graph is called the chromatic number of and denoted by In
this paper the concepts of '-chromatic sum and -chromatic sum are
introduced. The extended graph of a graph was recently introduced for
certain regular graphs. We further the concepts of '-chromatic sum and
-chromatic sum to extended paths and cycles. The paper concludes with
\emph{patterned structured} graphs.Comment: 12 page
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