51 research outputs found
Interval minors of complete bipartite graphs
Interval minors of bipartite graphs were recently introduced by Jacob Fox in
the study of Stanley-Wilf limits. We investigate the maximum number of edges in
-interval minor free bipartite graphs. We determine exact values when
and describe the extremal graphs. For , lower and upper bounds are
given and the structure of -interval minor free graphs is studied
Beyond Ohba's Conjecture: A bound on the choice number of -chromatic graphs with vertices
Let denote the choice number of a graph (also called "list
chromatic number" or "choosability" of ). Noel, Reed, and Wu proved the
conjecture of Ohba that when . We
extend this to a general upper bound: . Our result is sharp for
using Ohba's examples, and it improves the best-known
upper bound for .Comment: 14 page
Monochromatic -connected Subgraphs in 2-edge-colored Complete Graphs
Bollob\'{a}s and Gy\'{a}rf\'{a}s conjectured that for any with , every 2-edge-coloring of the complete graph on
vertices leads to a -connected monochromatic subgraph with at least
vertices. We find a counterexample with , thus disproving the conjecture, and we show the
conjecture is true for
The perturbation bound of the extended vertical linear complementarity problem
In this paper, we discuss the perturbation analysis of the extended vertical
linear complementarity problem (EVLCP). Under the assumption of the row
-property, several absolute and relative perturbation bounds of
EVLCP are given, which can be reduced to some existing results. Some numerical
examples are given to show the proposed bounds
Large Supports are required for Well-Supported Nash Equilibria
We prove that for any constant and any , there exist bimatrix
win-lose games for which every -WSNE requires supports of cardinality
greater than . To do this, we provide a graph-theoretic characterization of
win-lose games that possess -WSNE with constant cardinality supports.
We then apply a result in additive number theory of Haight to construct
win-lose games that do not satisfy the requirements of the characterization.
These constructions disprove graph theoretic conjectures of Daskalakis, Mehta
and Papadimitriou, and Myers
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