scholarly journals Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1755
Author(s):  
Giovanni Lo Faro ◽  
Salvatore Milici ◽  
Antoinette Tripodi
Keyword(s):  
Connected Graph ◽  
Existence Problem ◽  
Spanning Subgraph ◽  
Isomorphic Graphs ◽  
Edge Decomposition

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.

Some results about component factors in graphs

2019 ◽  
Vol 53 (3) ◽  
pp. 723-730 ◽  
Author(s):  
Sizhong Zhou
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Path Factor

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P≥ k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k − 1). A graph G is called a P≥ k-factor covered graph, if for each edge e of G, there is a P≥ k-factor covering e. In this paper, we prove that (1) a graph G has a {K1,1,K1,2, … ,K1,k}-factor if and only if bind(G) ≥ 1/k, where k ≥ 2 is an integer; (2) a connected graph G is a P≥ 2-factor covered graph if bind(G) > 2/3; (3) a connected graph G is a P≥ 3-factor covered graph if bind(G) ≥ 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.

5-Shredders of Contraction-Critical 5-Connected Graphs

2020 ◽  
Vol 30 (03) ◽  
pp. 2040008
Author(s):  
Chengfu Qin ◽  
Weihua Yang
Keyword(s):  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph

Yoshimi Egawa [8] showed that a 5-connected graph G admits at most [Formula: see text] 5-shredders. In this paper we shown that a contraction-critical 5-connected graph G admits at most [Formula: see text] 5-shredders. Further we show that, for every contraction-critical 5-connected graph G, there is a contraction critical 5-connected graph [Formula: see text] such that G is a spanning subgraph of [Formula: see text] and [Formula: see text] admits at most [Formula: see text] 5-shredders.

Component factors of the Cartesian product of graphs

2016 ◽  
Vol 09 (02) ◽  
pp. 1650041
Author(s):  
M. R. Chithra ◽  
A. Vijayakumar
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Induced Subgraph ◽  
Connected Graphs ◽  
Cartesian Product Of Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Product Of Graphs

Let [Formula: see text] be a family of connected graphs. A spanning subgraph [Formula: see text] of [Formula: see text] is called an [Formula: see text]-factor (component factor) of [Formula: see text] if each component of [Formula: see text] is in [Formula: see text]. In this paper, we study the component factors of the Cartesian product of graphs. Here, we take [Formula: see text] and show that every connected graph [Formula: see text] has a [Formula: see text]-factor where [Formula: see text] and [Formula: see text] is the maximum degree of an induced subgraph [Formula: see text] in [Formula: see text] or [Formula: see text]. Also, we characterize graphs [Formula: see text] having a [Formula: see text]-factor.

scholarly journals Isolated toughness and path-factor uniform graphs

2021 ◽  
Author(s):  
Sizhong Zhou ◽  
Zhiren Sun ◽  
Hongxia Liu
Keyword(s):  
Connected Graph ◽  
Edge Connectivity ◽  
Spanning Subgraph ◽  
K Factor ◽  
Path Factor

A $P_{\geq k}$-factor of a graph $G$ is a spanning subgraph of $G$ whose components are paths of order at least $k$. We say that a graph $G$ is $P_{\geq k}$-factor covered if for every edge $e\in E(G)$, $G$ admits a $P_{\geq k}$-factor that contains $e$; and we say that a graph $G$ is $P_{\geq k}$-factor uniform if for every edge $e\in E(G)$, the graph $G-e$ is $P_{\geq k}$-factor covered. In other words, $G$ is $P_{\geq k}$-factor uniform if for every pair of edges $e_1,e_2\in E(G)$, $G$ admits a $P_{\geq k}$-factor that contains $e_1$ and avoids $e_2$. In this article, we testify that (\romannumeral1) a 3-edge-connected graph $G$ is $P_{\geq2}$-factor uniform if its isolated toughness $I(G)>1$; (\romannumeral2) a 3-edge-connected graph $G$ is $P_{\geq3}$-factor uniform if its isolated toughness $I(G)>2$. Furthermore, we explain that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.

CONSTRUCTING MULTIDIMENSIONAL SPANNER GRAPHS

1991 ◽  
Vol 01 (02) ◽  
pp. 99-107 ◽  
Author(s):  
JEFFERY S. SALOWE
Keyword(s):  
Shortest Path ◽  
Complete Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Input Parameter ◽  
Minimum Spanning Trees ◽  
Positive Edge ◽  
Spanning Subgraph ◽  
Vertex Set

Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in [Formula: see text] and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing [Formula: see text] edges in [Formula: see text] time. We apply this spanner to the construction of approximate minimum spanning trees.

scholarly journals On a Class of Polynomials Associated with the Stars of a Graph and its Application to Node-Disjoint Decompositions of Complete Graphs and Complete Bipartite Graphs into Stars

1979 ◽  
Vol 22 (1) ◽  
pp. 35-46 ◽  
Author(s):  
E. J. Farrell
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Complete Bipartite Graphs

AbstractA star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.

A linear-time algorithm for finding a sparsek-connected spanning subgraph of ak-connected graph

Algorithmica ◽  
1992 ◽  
Vol 7 (1-6) ◽  
pp. 583-596 ◽  
Author(s):  
Hiroshi Nagamochi ◽  
Toshihide Ibaraki
Keyword(s):  
Connected Graph ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Spanning Subgraph

scholarly journals Component factors and binding number conditions in graphs

2021 ◽  
Vol 6 (11) ◽  
pp. 12460-12470
Author(s):  
Sizhong Zhou ◽  
◽  
Jiang Xu ◽  
Lan Xu ◽  
Keyword(s):  
Connected Graph ◽  
Connected Graphs ◽  
Binding Number ◽  
Spanning Subgraph ◽  
Critical Graph

<abstract><p>Let $ G $ be a graph. For a set $ \mathcal{H} $ of connected graphs, an $ \mathcal{H} $-factor of a graph $ G $ is a spanning subgraph $ H $ of $ G $ such that every component of $ H $ is isomorphic to a member of $ \mathcal{H} $. A graph $ G $ is called an $ (\mathcal{H}, m) $-factor deleted graph if for every $ E'\subseteq E(G) $ with $ |E'| = m $, $ G-E' $ admits an $ \mathcal{H} $-factor. A graph $ G $ is called an $ (\mathcal{H}, n) $-factor critical graph if for every $ N\subseteq V(G) $ with $ |N| = n $, $ G-N $ admits an $ \mathcal{H} $-factor. Let $ m $, $ n $ and $ k $ be three nonnegative integers with $ k\geq2 $, and write $ \mathcal{F} = \{P_2, C_3, P_5, \mathcal{T}(3)\} $ and $ \mathcal{H} = \{K_{1, 1}, K_{1, 2}, \cdots, K_{1, k}, \mathcal{T}(2k+1)\} $, where $ \mathcal{T}(3) $ and $ \mathcal{T}(2k+1) $ are two special families of trees. In this article, we verify that (i) a $ (2m+1) $-connected graph $ G $ is an $ (\mathcal{F}, m) $-factor deleted graph if its binding number $ bind(G)\geq\frac{4m+2}{2m+3} $; (ii) an $ (n+2) $-connected graph $ G $ is an $ (\mathcal{F}, n) $-factor critical graph if its binding number $ bind(G)\geq\frac{2+n}{3} $; (iii) a $ (2m+1) $-connected graph $ G $ is an $ (\mathcal{H}, m) $-factor deleted graph if its binding number $ bind(G)\geq\frac{2}{2k-1} $; (iv) an $ (n+2) $-connected graph $ G $ is an $ (\mathcal{H}, n) $-factor critical graph if its binding number $ bind(G)\geq\frac{2+n}{2k+1} $.</p></abstract>

scholarly journals Edge-Decomposition of Graphs into Copies of a Tree with Four Edges

10.37236/2110 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
János Barát ◽  
Dániel Gerbner
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Degree Sequence ◽  
Decomposition Theorem ◽  
Bipartite Graphs ◽  
Connected Graphs ◽  
Edge Decomposition

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013).Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant.

ON FINDING SPARSE THREE-EDGE-CONNECTED AND THREE-VERTEX-CONNECTED SPANNING SUBGRAPHS

2014 ◽  
Vol 25 (03) ◽  
pp. 355-368
Author(s):  
AMR ELMASRY ◽  
YUNG H. TSIN
Keyword(s):  
Connected Graph ◽  
Sparse Graph ◽  
Input Graph ◽  
Vertex Connectivity ◽  
Time Efficiency ◽  
Depth First Search ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Ear Decomposition ◽  
The Given

We present algorithms that construct a sparse spanning subgraph of a three-edge-connected graph that preserves three-edge connectivity or of a three-vertex-connected graph that preserves three-vertex connectivity. Our algorithms are conceptually simple and run in O(|E|) time. These simple algorithms can be used to improve the efficiency of the best-known algorithms for three-edge and three-vertex connectivity and their related problems, by preprocessing the input graph so as to trim it down to a sparse graph. Afterwards, the original algorithms run in O(|V|) instead of O(|E|) time. Our algorithms generate an adjacency-lists structure to represent the sparse spanning subgraph, so that when a depth-first search is performed over the subgraph based on this adjacency-lists structure it actually traverses the paths in an ear-decomposition of the subgraph. This is useful because many of the existing algorithms for three-edge- or three-vertex connectivity and their related problems are based on an ear-decomposition of the given graph. Using such an adjacency-lists structure to represent the input graph would greatly improve the run-time efficiency of these algorithms.

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