scholarly journals On end-faithful spanning trees in infinite graphs

1990 ◽  
Vol 107 (3) ◽  
pp. 461-473 ◽  
Author(s):  
Reinhard Diestel
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Infinite Graphs ◽  
Connected Subgraph ◽  
Initial Vertex ◽  
Infinite Path ◽  
Infinite Component

Let G be an infinite connected graph. A ray (from ν) in G is a 1-way infinite path in G (with initial vertex ν). An infinite connected subgraph of a ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X.

scholarly journals Rainbow Paths with Prescribed Ends

10.37236/573 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Meysam Alishahi ◽  
Ali Taherkhani ◽  
Carsten Thomassen
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Vertex Coloring ◽  
Initial Vertex ◽  
Locally Finite ◽  
Circular Chromatic Number ◽  
Connected Graphs ◽  
Infinite Path

It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if $G$ is a connected graph distinct from $C_7$, then there is a $\chi(G)$-coloring of $G$ in which every vertex $v\in V(G)$ is an initial vertex of a path $P$ with $\chi(G)$ vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs. Electron. J. Combin. 18(1): P17, 9pp, 2011] this was proved with $\lfloor\frac{\chi(G)}{2} \rfloor $ vertices instead of $\chi(G)$ vertices. We strengthen this to $\chi(G)-1$ vertices. We also prove that every connected graph with at least one edge has a proper $k$-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color $i+1$ (mod $k$). $C_5$ shows that $k$ may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the $k$-coloring exists for every $k \geq \chi(G)$. In fact, the $k$-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by $1$ (mod $k$) along each edge. The method is based on the circular chromatic number $\chi_c(G)$. In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number.

scholarly journals Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

10.37236/3046 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Simon M. Smith ◽  
Mark E. Watkins
Keyword(s):  
Connected Graph ◽  
Automorphism Groups ◽  
Cardinal Number ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Finite Radius ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Identity Permutation ◽  
Imprimitive Group

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

A Note on Primitive Graphs

1972 ◽  
Vol 15 (3) ◽  
pp. 437-440 ◽  
Author(s):  
I. Z. Bouwer ◽  
G. F. LeBlanc
Keyword(s):  
Connected Graph ◽  
Common Vertex ◽  
Connected Subgraph ◽  
Property A ◽  
Vertex Set

Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

scholarly journals Topological paths, cycles and spanning trees in infinite graphs

2004 ◽  
Vol 25 (6) ◽  
pp. 835-862 ◽  
Author(s):  
Reinhard Diestel ◽  
Daniela Kühn
Keyword(s):  
Spanning Trees ◽  
Infinite Graphs

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

scholarly journals Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review

2018 ◽  
Vol 5 (3) ◽  
pp. 265-281 ◽  
Author(s):  
Maumita Chakraborty ◽  
Sumon Chowdhury ◽  
Joymallya Chakraborty ◽  
Ranjan Mehera ◽  
Rajat Kumar Pal
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Extensive Review
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