744 research outputs found
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
Academic team formation as evolving hypergraphs
This paper quantitatively explores the social and socio-semantic patterns of
constitution of academic collaboration teams. To this end, we broadly underline
two critical features of social networks of knowledge-based collaboration:
first, they essentially consist of group-level interactions which call for
team-centered approaches. Formally, this induces the use of hypergraphs and
n-adic interactions, rather than traditional dyadic frameworks of interaction
such as graphs, binding only pairs of agents. Second, we advocate the joint
consideration of structural and semantic features, as collaborations are
allegedly constrained by both of them. Considering these provisions, we propose
a framework which principally enables us to empirically test a series of
hypotheses related to academic team formation patterns. In particular, we
exhibit and characterize the influence of an implicit group structure driving
recurrent team formation processes. On the whole, innovative production does
not appear to be correlated with more original teams, while a polarization
appears between groups composed of experts only or non-experts only, altogether
corresponding to collectives with a high rate of repeated interactions
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