216 research outputs found
On the multipacking number of grid graphs
In 2001, Erwin introduced broadcast domination in graphs. It is a variant of
classical domination where selected vertices may have different domination
powers. The minimum cost of a dominating broadcast in a graph is denoted
. The dual of this problem is called multipacking: a multipacking
is a set of vertices such that for any vertex and any positive integer
, the ball of radius around contains at most vertices of .
The maximum size of a multipacking in a graph is denoted mp(G). Naturally
mp(G) . Earlier results by Farber and by Lubiw show that
broadcast and multipacking numbers are equal for strongly chordal graphs. In
this paper, we show that all large grids (height at least 4 and width at least
7), which are far from being chordal, have their broadcast and multipacking
numbers equal
A Dichotomy Theorem for Circular Colouring Reconfiguration
The "reconfiguration problem" for circular colourings asks, given two
-colourings and of a graph , is it possible to transform
into by changing the colour of one vertex at a time such that every
intermediate mapping is a -colouring? We show that this problem can be
solved in polynomial time for and is PSPACE-complete for
. This generalizes a known dichotomy theorem for reconfiguring
classical graph colourings.Comment: 22 pages, 5 figure
Building blocks for the variety of absolute retracts
AbstractGiven a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k⩾2 we describe a collection of graphs that generate the variety ARk of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that ARk⊂NUFk+1, where NUFk+1 is the variety of graphs that admit a near unanimity function of arity k+1
Characterizing Circular Colouring Mixing for
Given a graph , the -mixing problem asks: Can one obtain all
-colourings of , starting from one -colouring , by changing the
colour of only one vertex at a time, while at each step maintaining a
-colouring? More generally, for a graph , the -mixing problem asks:
Can one obtain all homomorphisms , starting from one homomorphism ,
by changing the image of only one vertex at a time, while at each step
maintaining a homomorphism ?
This paper focuses on a generalization of -colourings, namely
-circular colourings. We show that when , a graph
is -mixing if and only if for any -colouring of , and
any cycle of , the wind of the cycle under the colouring equals a
particular value (which intuitively corresponds to having no wind). As a
consequence we show that -mixing is closed under a restricted
homomorphism called a fold. Using this, we deduce that -mixing is
co-NP-complete for all , and by similar ideas we show that if
the circular chromatic number of a connected graph is ,
then folds to . We use the characterization to settle a
conjecture of Brewster and Noel, specifically that the circular mixing number
of bipartite graphs is . Lastly, we give a polynomial time algorithm for
-mixing in planar graphs when .Comment: 21 page
Extending precolourings of circular cliques
Let G be a graph with circular chromatic number Xe(G) = k/q. Given P ⊆ V (G) where the components of G [P] are isomorphic to the circular clique Gk,g suppose the vertices of P have been precoloured with a (k’,g’) - colouring. We examine under what conditions one can be assured the colouring extends to the entire graph. We stud y sufficient conditions based on k’/q’ − k/q as well as the distance between precoloured components of G[P]. In particular, we examine a conjecture of Albertson and West showing the conditions for extendibility are more complex than anticipated in their work
Halomethyl-Triazoles for Rapid, Site-Selective Protein Modification
Post-translational modifications (PTMs) are used by organisms to control protein structure and function after protein translation, but their study is complicated and their roles are not often well understood as PTMs are difficult to introduce onto proteins selectively. Designing reagents that are both good mimics of PTMs, but also only modify select amino acid residues in proteins is challenging. Frequently, both a chemical warhead and linker are used, creating a product that is a misrepresentation of the natural modification. We have previously shown that biotin-chloromethyl-triazole is an effective reagent for cysteine modification to give S-Lys derivatives where the triazole is a good mimic of natural lysine acylation. Here, we demonstrate both how the reactivity of the alkylating reagents can be increased and how the range of triazole PTM mimics can be expanded. These new iodomethyl-triazole reagents are able to modify a cysteine residue on a histone protein with excellent selectivity in 30 min to give PTM mimics of acylated lysine side-chains. Studies on the more complicated, folded protein SCP-2L showed promising reactivity, but also suggested the halomethyl-triazoles are potent alkylators of methionine residues
New Research on the Cowling and Cooling of Radial Engines
An extensive series of wind-tunnel tests on a half-scale conventional, nacelle model were made by the United Aircraft Corporation to determine and correlate the effects of many variables on cooling air flow and nacelle drag. The primary investigation was concerned with the reaction of these factors to varying conditions ahead of, across, and behind the engine. In the light of this investigation, common misconceptions and factors which are frequently overlooked in the cooling and cowling of radial engines are considered in some detail. Data are presented to support certain design recommendations and conclusions which should lead toward the improvement of present engine installations. Several charts are included to facilitate the estimation of cooling drag, available cooling pressure, and cowl exit area
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