scholarly journals Permanent Index of Matrices Associated with Graphs

10.37236/5494 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Tsai-Lien Wong ◽  
Xuding Zhu
Keyword(s):  
Graph Theory ◽  
Real Number ◽  
Reduction Method ◽  
Total Weight ◽  
Combinatorial Nullstellensatz ◽  
Induced Subgraphs ◽  
Halin Graph ◽  
Subcubic Graph ◽  
Promising Tool ◽  
Total Weighting

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $\phi_f(u) \ne \phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz.  This approach leads to conjectures on the permanent indices of matrices  $A_G$ and $B_G$ associated to a graph $G$. In this  paper, we establish a method that reduces the study of permanent of  matrices associated to a graph $G$ to the study of permanent of  matrices associated to induced subgraphs of $G$. Using this  reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable.

scholarly journals A complete classification of the complexity of the vertex 3-colourability problem for quadruples of induced 5-vertex prohibitions

2020 ◽  
Vol 22 (1) ◽  
pp. 38-47
Author(s):  
Dmitry S. Malyshev
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complete Classification ◽  
Induced Subgraphs ◽  
Simple Graphs ◽  
Forbidden Induced Subgraphs ◽  
The Third ◽  
Hereditary Class ◽  
Complexity Status

The vertex 3-colourability problem for a given graph is to check whether it is possible to split the set of its vertices into three subsets of pairwise non-adjacent vertices or not. A hereditary class of graphs is a set of simple graphs closed under isomorphism and deletion of vertices; the set of its forbidden induced subgraphs defines every such a class. For all but three the quadruples of 5-vertex forbidden induced subgraphs, we know the complexity status of the vertex 3-colourability problem. Additionally, two of these three cases are polynomially equivalent; they also polynomially reduce to the third one. In this paper, we prove that the computational complexity of the considered problem in all of the three mentioned classes is polynomial. This result contributes to the algorithmic graph theory.

Expected Maximum Block Size in Critical Random Graphs

2019 ◽  
Vol 28 (4) ◽  
pp. 638-655
Author(s):  
V. Rasendrahasina ◽  
A. Rasoanaivo ◽  
V. Ravelomanana
Keyword(s):  
Real Number ◽  
Random Graphs ◽  
Function Theory ◽  
Block Size ◽  
Maximum Size ◽  
Induced Subgraphs ◽  
Symbolic Method ◽  
Exact Analytic Expression ◽  
Analytic Expression ◽  
Image Position

AbstractLet G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that $$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$ Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for $$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$ This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$ as a function of λ.

scholarly journals Lai's Conditions for Spanning and Dominating Closed Trails

10.37236/4511 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Wei-Guo Chen ◽  
Zhi-Hong Chen ◽  
Mei Lu
Keyword(s):  
Graph Theory ◽  
Real Number ◽  
Connected Graph ◽  
Finite Family ◽  
Discrete Mathematics ◽  
Line Graphs ◽  
Connected Graphs ◽  
Supereulerian Graphs ◽  
The Family ◽  
Hamiltonian Line Graphs

A graph is supereulerian if it has a spanning closed trail. For an integer $r$, let ${\cal Q}_0(r)$ be  the family of 3-edge-connected nonsupereulerian graphs of order at most $r$. For a graph $G$, define $\delta_L(G)=\min\{\max\{d(u), d(v) \}| \  \mbox{ for any $uv\in E(G)$} \}$. For a given integer $p\ge 2$ and a given real number $\epsilon$,  a graph $G$ of order $n$ is said to satisfy a Lai's condition if $\delta_L(G)\ge \frac{n}{p}-\epsilon$.  In this paper, we show that  if $G$ is  a  3-edge-connected graph of order $n$ with $\delta_L(G)\ge \frac{n}{p}-\epsilon$, then there is an integer $N(p, \epsilon)$ such that when $n> N(p,\epsilon)$, $G$ is supereulerian if and only if $G$ is not  a graph obtained from a  graph $G_p$ in the finite family ${\cal Q}_0(3p-5)$ by replacing some vertices in $G_p$ with nontrivial graphs. Results on the best possible Lai's  conditions for Hamiltonian line graphs of 3-edge-connected graphs or 3-edge-connected supereulerian graphs are given,  which are improvements of the results in [J. Graph Theory 42(2003) 308-319] and in [Discrete Mathematics, 310(2010) 2455-2459].

scholarly journals A new characterization and a recognition algorithm of Lucas cubes

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Andrej Taranenko
Keyword(s):  
Graph Theory ◽  
Interconnection Networks ◽  
Linear Time ◽  
Recognition Algorithm ◽  
Induced Subgraphs ◽  
Fibonacci Cube ◽  
Vertex Set ◽  
International Audience ◽  
Binary Strings

Graph Theory International audience Fibonacci and Lucas cubes are induced subgraphs of hypercubes obtained by excluding certain binary strings from the vertex set. They appear as models for interconnection networks, as well as in chemistry. We derive a characterization of Lucas cubes that is based on a peripheral expansion of a unique convex subgraph of an appropriate Fibonacci cube. This serves as the foundation for a recognition algorithm of Lucas cubes that runs in linear time.

scholarly journals Extremal $H$-Free Planar Graphs

10.37236/8255 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Yongxin Lan ◽  
Yongtang Shi ◽  
Zi-Xia Song
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Sufficient Conditions ◽  
Subcubic Graph ◽  
Stone Theorem

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory  83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield  $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem.  We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to  $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

scholarly journals On probe 2-clique graphs and probe diamond-free graphs

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Flavia Bonomo ◽  
Celina M. H. Figueiredo ◽  
Guillermo Duran ◽  
Luciano N. Grippo ◽  
Martín D. Safe ◽  
...  
Keyword(s):  
Graph Theory ◽  
Recognition Algorithm ◽  
Chordal Graphs ◽  
Stable Set ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Block Graphs ◽  
International Audience ◽  
Clique Graphs ◽  
Probe Block

Graph Theory International audience Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.

scholarly journals Total Weight Choosability of Graphs

10.37236/599 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Main Tool ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Total Weight ◽  
Combinatorial Nullstellensatz ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Proper Vertex ◽  
Vertex Colouring

Suppose the edges and the vertices of a simple graph $G$ are assigned $k$-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there any finite bound or maybe already lists of length two are sufficient? We prove that $2$-element lists are enough for trees, wheels, unicyclic and complete graphs, while the ones of length $3$ are sufficient for complete bipartite graphs. Our main tool is an algebraic theorem by Alon called Combinatorial Nullstellensatz.

scholarly journals Combinatorial Nullstellensatz

1999 ◽  
Vol 8 (1-2) ◽  
pp. 7-29 ◽  
Author(s):  
NOGA ALON
Keyword(s):  
Number Theory ◽  
Graph Theory ◽  
Additive Number Theory ◽  
Graph Colouring ◽  
Combinatorial Nullstellensatz ◽  
Additive Number ◽  
Combinatorial Number Theory ◽  
Algebraic Technique

We present a general algebraic technique and discuss some of its numerous applications in combinatorial number theory, in graph theory and in combinatorics. These applications include results in additive number theory and in the study of graph colouring problems. Many of these are known results, to which we present unified proofs, and some results are new.

scholarly journals Improved bounds for the Erdős-Rogers function

2020 ◽  
Author(s):  
Timothy Gowers ◽  
Oliver Janzer
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Related Problem ◽  
Independent Set ◽  
Large Structure ◽  
Induced Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Subgraph

[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{\alpha_s}$ vertices with $\alpha_s<1/2$ for every $s\ge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2\le t\le 2s-1$ and $s\ge 3$.

Practical applications of scanning tunneling microscopy

1989 ◽  
Vol 47 ◽  
pp. 18-19
Author(s):  
D. R. Denley
Keyword(s):  
High Resolution ◽  
Scanning Tunneling Microscopy ◽  
Quantitative Information ◽  
Digital Data ◽  
Scanning Tunneling ◽  
Ambient Atmosphere ◽  
Tunneling Microscopy ◽  
Clear Trend ◽  
Practical Applications ◽  
Promising Tool

Scanning tunneling microscopy (STM) has recently been introduced as a promising tool for analyzing surface atomic structure. We have used STM for its extremely high resolution (especially the direction normal to surfaces) and its ability for imaging in ambient atmosphere. We have examined surfaces of metals, semiconductors, and molecules deposited on these materials to achieve atomic resolution in favorable cases.When the high resolution capability is coupled with digital data acquisition, it is simple to get quantitative information on surface texture. This is illustrated for the measurement of surface roughness of evaporated gold films as a function of deposition temperature and annealing time in Figure 1. These results show a clear trend for which the roughness, as well as the experimental deviance of the roughness is found to be minimal for evaporation at 300°C. It is also possible to contrast different measures of roughness.

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