scholarly journals Every Graph $G$ is Hall $\Delta(G)$-Extendible

10.37236/5687 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Sarah Holliday ◽  
Jennifer Vandenbussche ◽  
Erik E. Westlund
Keyword(s):  
Independence Number ◽  
Affirmative Answer ◽  
List Coloring ◽  
Complete Spectrum ◽  
Odd Cycle ◽  
Graph Families ◽  
Delta G ◽  
Hall's Condition

In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.

scholarly journals Completing Partial Proper Colorings using Hall's Condition

10.37236/4387 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Sarah Holliday ◽  
Jennifer Vandenbussche ◽  
Erik E Westlund
Keyword(s):  
Planar Graph ◽  
Independence Number ◽  
List Coloring ◽  
Proper Coloring ◽  
Odd Cycle ◽  
Delta G ◽  
Hall's Condition

In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph $G$ with list assignment $L$ satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $m$-completable for some $m \geq \chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \geq 3$? (2) If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-completable? This paper establishes that for every $m \geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\Delta(G)$-completable.

scholarly journals On the Independence Numbers of the Cubes of Odd Cycles

10.37236/2598 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Tom Bohman ◽  
Ron Holzman ◽  
Venkatesh Natarajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Independence Number ◽  
Odd Cycle ◽  
Odd Cycles ◽  
Independence Numbers

We give an upper bound on the independence number of the cube of the odd cycle $C_{8n+5}$. The best known lower bound is conjectured to be the truth; we prove the conjecture in the case $8n+5$ prime and, within $2$, for general $n$.

Restricted list coloring and Hall's condition

2002 ◽  
Vol 249 (1-3) ◽  
pp. 57-63
Author(s):  
M.M. Cropper ◽  
J.L. Goldwasser
Keyword(s):  
List Coloring ◽  
Hall's Condition

scholarly journals Beyond Degree Choosability

10.37236/6179 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
List Coloring ◽  
Odd Cycle ◽  
Eulerian Subgraphs

Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem states that $G$ has a $\Delta$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is degree-choosable if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a Gallai tree. This degree-choosability result was further strengthened to Alon—Tarsi orientations; these are orientations of $G$ in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph $G$ is degree-AT if $G$ has an Alon—Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also degree-choosable. Hladký, Král', and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such that $G$ has no Alon—Tarsi orientation in which each vertex has indegree at least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the characterization is simple to state.

scholarly journals The Biased Odd Cycle Game

10.37236/2819 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Asaf Ferber ◽  
Roman Glebov ◽  
Michael Krivelevich ◽  
Hong Liu ◽  
Cory Palmer ◽  
...  
Keyword(s):  
Difficult Problem ◽  
Odd Cycle ◽  
Delta G ◽  
Special Case

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $\delta>0$ and large enough $n$, there exists a constant $k$ for which if $\delta(G)\geq \delta n$ and $\chi(G)\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O\left(\frac{n}{\log^2 n}\right)$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, Łuczak and Rödl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $\chi(G)\geq k$, then Maker can build a graph which is not $t$-colorable, in the $(1:b)$ Maker-Breaker game played on the vertices of $G$?

scholarly journals On the Alon-Tarsi Number and Chromatic-Choosability of Cartesian Products of Graphs

10.37236/7740 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Hemanshu Kaul ◽  
Jeffrey A. Mudrock
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Hamiltonian Path ◽  
Complete Graphs ◽  
Cartesian Products ◽  
Hamilton Path ◽  
Odd Cycle ◽  
Delta G ◽  
List Chromatic Number ◽  
Cartesian Products Of Graphs

We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $\chi(G) \leq \chi_\ell(G) \leq \chi_p(G) \leq AT(G)$, where  $\chi(G)$ is the chromatic number, $\chi_\ell(G)$ is the list chromatic number, and $\chi_p(G)$ is the paint number of $G$. In this paper we find families of graphs $G$ and $H$ such that $\chi(G \square H) = AT(G \square H)$, reducing this sequence of inequalities to equality. We show that the Alon-Tarsi number of the Cartesian product of an odd cycle and a path is always equal to 3. This result is then extended to show that if $G$ is an odd cycle or a complete graph and $H$ is a graph on at least two vertices containing the Hamilton path $w_1, w_2, \ldots, w_n$ such that for each $i$, $w_i$ has a most $k$ neighbors among $w_1, w_2, \ldots, w_{i-1}$, then $AT(G \square H) \leq \Delta(G)+k$ where $\Delta(G)$ is the maximum degree of $G$.  We discuss other extensions for $G \square H$, where $G$ is such that $V(G)$ can be partitioned into odd cycles and complete graphs, and $H$ is a graph containing a Hamiltonian path. We apply these bounds to get chromatic-choosable Cartesian products, in fact we show that these families of graphs have $\chi(G) = AT(G)$, improving previously known bounds.

scholarly journals Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability

10.37236/7163 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Thomas Mahoney
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Online Version ◽  
Independent Subset ◽  
Odd Cycle ◽  
Disjoint Sets ◽  
Delta G ◽  
To Receive

Let $a,b\in\mathbb{N}$. A graph $G$ is $(a,b)$-choosable if for any list assignment $L$ such that $|L(v)|\ge a$, there exists a coloring in which each vertex $v$ receives a set $C(v)$ of $b$ colors such that $C(v)\subseteq L(v)$ and $C(u)\cap C(w)=\emptyset$ for any $uw\in E(G)$. In the online version of this problem, on each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of these vertices to receive that color. We say $G$ is $(a,b)$-paintable if when each vertex $v$ is allowed to be marked $a$ times, there is an algorithm to produce a coloring in which each vertex $v$ receives $b$ colors such that adjacent vertices receive disjoint sets of colors.We show that every odd cycle $C_{2k+1}$ is $(a,b)$-paintable exactly when it is $(a,b)$-chosable, which is when $a\ge2b+\lceil b/k\rceil$. In 2009, Zhu conjectured that if $G$ is $(a,1)$-paintable, then $G$ is $(am,m)$-paintable for any $m\in\mathbb{N}$. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any $m \in \mathbb{N}$ that $G$ is $(5m,m)$-paintable when $G$ is planar. Strengthening work of Tuza and Voigt, and of Hladky, Kral, and Schauz, we prove that for any connected graph $G$ other than an odd cycle or complete graph and any $m\in\mathbb{N}$, $G$ is $(\Delta(G)m,m)$-paintable.

Confocal Raman mapping

1992 ◽  
Vol 50 (2) ◽  
pp. 1514-1515
Author(s):  
J. Barbillat ◽  
M. Delhaye ◽  
P. Dhamelincourt
Keyword(s):  
Chemical Species ◽  
Diffraction Limit ◽  
Microscopic Level ◽  
Raman Mapping ◽  
Complete Spectrum ◽  
Laser Illumination ◽  
Ccd Detector ◽  
Raman Microprobe ◽  
Photodiode Array ◽  
Raman Image

Raman mapping, with a spatial resolution close to the diffraction limit, can help to reveal the distribution of chemical species at the surface of an heterogeneous sample.As early as 1975,three methods of sample laser illumination and detector configuration have been proposed to perform Raman mapping at the microscopic level (Fig. 1),:- Point illumination:The basic design of the instrument is a classical Raman microprobe equipped with a PM tube or either a linear photodiode array or a two-dimensional CCD detector. A laser beam is focused on a very small area ,close to the diffraction limit.In order to explore the whole surface of the sample,the specimen is moved sequentially beneath the microscope by means of a motorized XY stage. For each point analyzed, a complete spectrum is obtained from which spectral information of interest is extracted for Raman image reconstruction.- Line illuminationA narrow laser line is focused onto the sample either by a cylindrical lens or by a scanning device and is optically conjugated with the entrance slit of the stigmatic spectrograph.

Quantitative Ranking of Ligand Binding Kinetics with a Multiscale Milestoning Simulation Approach

2018 ◽  
Author(s):  
Benjamin R. Jagger ◽  
Christoper T. Lee ◽  
Rommie Amaro
Keyword(s):  
Molecular Dynamics ◽  
Drug Discovery ◽  
Small Molecule ◽  
Brownian Dynamics ◽  
Model Simulation ◽  
Computational Cost ◽  
Lead Selection ◽  
Delta G ◽  
Dynamics Simulations

<p>The ranking of small molecule binders by their kinetic (kon and koff) and thermodynamic (delta G) properties can be a valuable metric for lead selection and optimization in a drug discovery campaign, as these quantities are often indicators of in vivo efficacy. Efficient and accurate predictions of these quantities can aid the in drug discovery effort, acting as a screening step. We have previously described a hybrid molecular dynamics, Brownian dynamics, and milestoning model, Simulation Enabled Estimation of Kinetic Rates (SEEKR), that can predict kon’s, koff’s, and G’s. Here we demonstrate the effectiveness of this approach for ranking a series of seven small molecule compounds for the model system, -cyclodextrin, based on predicted kon’s and koff’s. We compare our results using SEEKR to experimentally determined rates as well as rates calculated using long-timescale molecular dynamics simulations and show that SEEKR can effectively rank the compounds by koff and G with reduced computational cost. We also provide a discussion of convergence properties and sensitivities of calculations with SEEKR to establish “best practices” for its future use.</p>

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