6,066 research outputs found
An extremal problem on potentially -graphic sequences
A sequence is potentially graphical if it has a realization
containing a as a subgraph, where is a complete
3-partite graph with partition sizes . Let denote
the smallest degree sum such that every -term graphical sequence with
is potentially graphical. In
this paper, we prove that for
We conjecture that equality holds for We prove
that this conjecture is true for .Comment: 5 page
Upward-closed hereditary families in the dominance order
The majorization relation orders the degree sequences of simple graphs into
posets called dominance orders. As shown by Hammer et al. and Merris, the
degree sequences of threshold and split graphs form upward-closed sets within
the dominance orders they belong to, i.e., any degree sequence majorizing a
split or threshold sequence must itself be split or threshold, respectively.
Motivated by the fact that threshold graphs and split graphs have
characterizations in terms of forbidden induced subgraphs, we define a class
of graphs to be dominance monotone if whenever no realization of
contains an element as an induced subgraph, and majorizes
, then no realization of induces an element of . We present
conditions necessary for a set of graphs to be dominance monotone, and we
identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure
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