scholarly journals Minimum degree and the minimum size of K2t-saturated graphs

2007 ◽  
Vol 307 (9-10) ◽  
pp. 1108-1114 ◽  
Author(s):  
Ronald J. Gould ◽  
John R. Schmitt
Keyword(s):  
Minimum Degree ◽  
Minimum Size ◽  
Saturated Graphs

scholarly journals Immunoglobulin G from patients with heparin-induced thrombocytopenia binds to a complex of heparin and platelet factor 4

Blood ◽  
1994 ◽  
Vol 83 (11) ◽  
pp. 3232-3239 ◽  
Author(s):  
JG Kelton ◽  
JW Smith ◽  
TE Warkentin ◽  
CP Hayward ◽  
GA Denomme ◽  
...  
Keyword(s):  
Minimum Degree ◽  
Equilibrium Dialysis ◽  
Fluid Phase ◽  
Platelet Factor 4 ◽  
Minimum Size ◽  
Sulfated Polysaccharides ◽  
Platelet Factor ◽  
Platelet Surface ◽  
Heparin Induced Thrombocytopenia

Abstract Heparin-induced thrombocytopenia (HIT) is an important complication of heparin therapy. Although there is general agreement that platelet activation in vitro by the HIT IgG is mediated by the platelet Fc receptor, the interaction among the antibody, heparin, and platelet membrane components is uncertain and debated. In this report, we describe studies designed to address these interactions. We found, as others have noted, that a variety of other sulfated polysaccharides could substitute for heparin in the reaction. Using polysaccharides selected for both size and charge, we found that reactivity depended on two independent factors: a certain minimum degree of sulfation per saccharide unit and a certain minimum size. Hence, highly sulfated but small (< 1,000 daltons) polysaccharides were not reactive nor were large but poorly sulfated polysaccharides. The ability of HIT IgG to recognize heparin by itself was tested by Ouchterlony gel diffusion, ammonium sulfate and polyethylene glycol precipitation, and equilibrium dialysis. No technique demonstrated reactivity. However, when platelet releasate was added to heparin and HIT IgG, a 50-fold increase in binding of radio-labeled heparin to HIT IgG was observed. The releasate was then depleted of proteins capable of binding to heparin by immunoaffinity chromatography. Only platelet factor 4-immunodepleted releasate lost its reactivity with HIT IgG and heparin. Finally, to determine whether the reaction occurred on the surface of platelets or in the fluid phase, washed platelets were incubated with HIT IgG or heparin and after a wash step, heparin or HIT IgG was added, respectively. Reactivity was only noted when platelets were preincubated with heparin. Consistent with these observations was the demonstration of the presence of PF4 on platelets using flow cytometry. These studies indicate that heparin and other large, highly sulfated polysaccharides bind to PF4 to form a reactive antigen on the platelet surface. HIT IgG then binds to this complex with activation of platelets through the platelet Fc receptors.

Dominating a Family of Graphs with Small Connected Subgraphs

2000 ◽  
Vol 9 (4) ◽  
pp. 309-313 ◽  
Author(s):  
YAIR CARO ◽  
RAPHAEL YUSTER
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Minimum Size ◽  
Dominating Sets ◽  
Asymptotic Results ◽  
Connected Dominating Sets ◽  
Vertex Set ◽  
Connected Subgraphs

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

scholarly journals Triangles in Ks-saturated graphs with minimum degree t

2020 ◽  
Vol 07 (01) ◽  
pp. 1-24
Author(s):  
Craig Timmons ◽  
◽  
Benjamin Cole ◽  
Albert Curry ◽  
David Davini
Keyword(s):  
Minimum Degree ◽  
Saturated Graphs

scholarly journals Constructive Upper Bounds for Cycle-Saturated Graphs of Minimum Size

10.37236/1055 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ronald Gould ◽  
Tomasz Łuczak ◽  
John Schmitt
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Saturated Graphs ◽  
Minimum Number

A graph $G$ is said to be $C_l$-saturated if $G$ contains no cycle of length $l$, but for any edge in the complement of $G$ the graph $G+e$ does contain a cycle of length $l$. The minimum number of edges of a $C_l$-saturated graph was shown by Barefoot et al. to be between $n+c_1{n\over l}$ and $n+c_2{n\over l}$ for some positive constants $c_1$ and $c_2$. This confirmed a conjecture of Bollobás. Here we improve the value of $c_2$ for $l \geq 8$.

scholarly journals Cycle-saturated graphs of minimum size

1996 ◽  
Vol 150 (1-3) ◽  
pp. 31-48 ◽  
Author(s):  
C.A. Barefoot ◽  
L.H. Clark ◽  
R.C. Entringer ◽  
T.D. Porter ◽  
L.A. Székely ◽  
...  
Keyword(s):  
Minimum Size ◽  
Saturated Graphs

scholarly journals Immunoglobulin G from patients with heparin-induced thrombocytopenia binds to a complex of heparin and platelet factor 4

Blood ◽  
1994 ◽  
Vol 83 (11) ◽  
pp. 3232-3239 ◽  
Author(s):  
JG Kelton ◽  
JW Smith ◽  
TE Warkentin ◽  
CP Hayward ◽  
GA Denomme ◽  
...  
Keyword(s):  
Minimum Degree ◽  
Equilibrium Dialysis ◽  
Fluid Phase ◽  
Platelet Factor 4 ◽  
Minimum Size ◽  
Sulfated Polysaccharides ◽  
Platelet Factor ◽  
Platelet Surface ◽  
Heparin Induced Thrombocytopenia

Heparin-induced thrombocytopenia (HIT) is an important complication of heparin therapy. Although there is general agreement that platelet activation in vitro by the HIT IgG is mediated by the platelet Fc receptor, the interaction among the antibody, heparin, and platelet membrane components is uncertain and debated. In this report, we describe studies designed to address these interactions. We found, as others have noted, that a variety of other sulfated polysaccharides could substitute for heparin in the reaction. Using polysaccharides selected for both size and charge, we found that reactivity depended on two independent factors: a certain minimum degree of sulfation per saccharide unit and a certain minimum size. Hence, highly sulfated but small (< 1,000 daltons) polysaccharides were not reactive nor were large but poorly sulfated polysaccharides. The ability of HIT IgG to recognize heparin by itself was tested by Ouchterlony gel diffusion, ammonium sulfate and polyethylene glycol precipitation, and equilibrium dialysis. No technique demonstrated reactivity. However, when platelet releasate was added to heparin and HIT IgG, a 50-fold increase in binding of radio-labeled heparin to HIT IgG was observed. The releasate was then depleted of proteins capable of binding to heparin by immunoaffinity chromatography. Only platelet factor 4-immunodepleted releasate lost its reactivity with HIT IgG and heparin. Finally, to determine whether the reaction occurred on the surface of platelets or in the fluid phase, washed platelets were incubated with HIT IgG or heparin and after a wash step, heparin or HIT IgG was added, respectively. Reactivity was only noted when platelets were preincubated with heparin. Consistent with these observations was the demonstration of the presence of PF4 on platelets using flow cytometry. These studies indicate that heparin and other large, highly sulfated polysaccharides bind to PF4 to form a reactive antigen on the platelet surface. HIT IgG then binds to this complex with activation of platelets through the platelet Fc receptors.

scholarly journals Saturated Graphs of Prescribed Minimum Degree

2016 ◽  
Vol 26 (2) ◽  
pp. 201-207 ◽  
Author(s):  
A. NICHOLAS DAY
Keyword(s):  
Minimum Degree ◽  
Upper Bounds ◽  
Saturated Graphs ◽  
Minimum Number

A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function satt(n,p), defined to be the minimum number of edges that a Kp-saturated graph on n vertices can have if it has minimum degree at least t. We prove that satt(n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for satt(n,p).

scholarly journals Locating-Dominating Sets and Identifying Codes in Graphs of Girth at least 5

10.37236/4562 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Camino Balbuena ◽  
Florent Foucaud ◽  
Adriana Hansberg
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Upper Bounds ◽  
Disjoint Paths ◽  
Minimum Size ◽  
Dominating Sets ◽  
Identifying Codes ◽  
Identifying Code ◽  
Codes In Graphs ◽  
Vertex Disjoint

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meeting these bounds.

scholarly journals Bounds for Identifying Codes in Terms of Degree Parameters

10.37236/2036 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Florent Foucaud ◽  
Guillem Perarnau
Keyword(s):  
Large Class ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Clique Number ◽  
Minimum Size ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code ◽  
Sharp Estimates

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\gamma^{\text{ID}}(G)$ denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{d}+c$. We use probabilistic tools to show that for any $d\geq 3$, $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d)}$ holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d^{3})}$. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, $\gamma^{\text{ID}}(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$.

scholarly journals On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

10.37236/8857 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Zi-Xia Song ◽  
Jingmei Zhang
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Minimum Degree ◽  
Bootstrap Percolation ◽  
Critical Graphs ◽  
Critical Graph ◽  
The Family ◽  
Saturated Graphs ◽  
Minimum Number ◽  
Monochromatic Copy

Given an integer $r\geqslant 1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G \nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots, {H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory 11(1987), 191–196], we study the minimum number of edges over all $(K_t, \mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily $K_t$-saturated graph to prove that for all $t\geqslant4 $ and $k\geqslant\max\{6, t\}$, there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if $G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$ e(G)\geqslant \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2} \right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{4,5\}$ and $k\geqslant6$. The method we develop in this paper may shed some light on attacking Hanson and Toft's conjecture.

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