A theorem on connected graphs in which every edge belongs to a 1-factor

Charles H. C. Little

doi:10.1017/s144678870002913x

A theorem on connected graphs in which every edge belongs to a 1-factor

Journal of the Australian Mathematical Society ◽

10.1017/s144678870002913x ◽

1974 ◽

Vol 18 (4) ◽

pp. 450-452 ◽

Cited By ~ 11

Author(s):

Charles H. C. Little

Keyword(s):

Symmetric Difference ◽

Connected Graphs

In this paper, we consider factor covered graphs, which are defined basically as connected graphs in which every edge belongs to a 1-factor. The main theorem is that for any two edges e and e′ of a factor covered graph, there is a cycle C passing through e and e′ such that the edge set of C is the symmetric difference of two 1-factors.

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Forbidden Subgraphs Generating Almost the Same Sets

Combinatorics Probability Computing ◽

10.1017/s0963548313000254 ◽

2013 ◽

Vol 22 (5) ◽

pp. 733-748 ◽

Cited By ~ 1

Author(s):

SHINYA FUJITA ◽

MICHITAKA FURUYA ◽

KENTA OZEKI

Keyword(s):

Finite Number ◽

Symmetric Difference ◽

Forbidden Subgraphs ◽

Exceptional Set ◽

Induced Subgraph ◽

Connected Graphs ◽

Graph Property ◽

The Relationship

Let $\mathcal{H}$ be a set of connected graphs. A graph G is said to be $\mathcal{H}$-free if G does not contain any element of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}_{k}(\mathcal{H})$ be the set of k-connected $\mathcal{H}$-free graphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow a finite exceptional set of graphs. But if the symmetric difference of $\mathcal{F}_{k}(\mathcal{H}_{1})$ and $\mathcal{F}_{k}(\mathcal{H}_{2})$ is finite and we allow a finite number of exceptions, no graph property can distinguish them. Motivated by this observation, we study when we obtain a finite symmetric difference. In this paper, our main aim is the following. If $|\mathcal{H}|\leq 3$ and the symmetric difference of $\mathcal{F}_{1}(\{H\})$ and $\mathcal{F}_{1}(\mathcal{H})$ is finite, then either $H\in \mathcal{H}$ or $|\mathcal{H}|=3$ and H=C3. Furthermore, we prove that if the symmetric difference of $\mathcal{F}_{k}(\{H_{1}\})$ and $\mathcal{F}_{k}(\{H_{2}\})$ is finite, then H1=H2.

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Forbidden Pairs with a Common Graph Generating Almost the Same Sets

The Electronic Journal of Combinatorics ◽

10.37236/6190 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

Shuya Chiba ◽

Jun Fujisawa ◽

Michitaka Furuya ◽

Hironobu Ikarashi

Keyword(s):

Symmetric Difference ◽

Forbidden Subgraph ◽

Transitive Graph ◽

Forbidden Subgraphs ◽

Induced Subgraph ◽

Connected Graphs ◽

The Family ◽

Vertex Transitive ◽

Vertex Transitive Graph

Let $\mathcal{H}$ be a family of connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ does not contain any members of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}(\mathcal{H})$ be the family of connected $\mathcal{H}$-free graphs. In this context, the members of $\mathcal{H}$ are called forbidden subgraphs.In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let $H_{1},H_{2},H_{3}$ be connected graphs of order at least three, and suppose that $H_{1}$ is twin-less. If the symmetric difference of $\mathcal{F}(\{H_{1},H_{2}\})$ and $\mathcal{F}(\{H_{1},H_{3}\})$ is finite and the tuple $(H_{1};H_{2},H_{3})$ is non-trivial in a sense, then $H_{2}$ and $H_{3}$ are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733–748] concerning forbidden pairs.

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Lower bounds of forwarding indices of graph products

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v37i2.33784 ◽

2017 ◽

Vol 37 (2) ◽

pp. 107-114

Author(s):

Mohamed Amine Boutiche

Keyword(s):

Lower Bounds ◽

Cartesian Product ◽

Symmetric Difference ◽

Graph Products ◽

Connected Graphs ◽

Subdivision Graph

In this paper, we give lower bounds of vertex and edge forwarding indices for cartesian product, join, composition, disjunction and symmetric difference of graphs. Moreover, we derive further lower bounds for several operators on connected graphs, such as subdivision graph and total graphs.

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The Formula to Count The Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops

Journal of Physics Conference Series ◽

10.1088/1742-6596/1751/1/012023 ◽

2021 ◽

Vol 1751 ◽

pp. 012023

Author(s):

F C Puri ◽

Wamiliana ◽

M Usman ◽

Amanto ◽

M Ansori ◽

...

Keyword(s):

Connected Graphs

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On minimum algebraic connectivity of graphs whose complements are bicyclic

Open Mathematics ◽

10.1515/math-2019-0119 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1490-1502 ◽

Cited By ~ 1

Author(s):

Jia-Bao Liu ◽

Muhammad Javaid ◽

Mohsin Raza ◽

Naeem Saleem

Keyword(s):

Connected Graph ◽

Diffusion Processes ◽

Laplacian Matrix ◽

Group Differences ◽

Algebraic Connectivity ◽

Connected Graphs ◽

Bicyclic Graph ◽

Smallest Eigenvalue ◽

Unique Graph ◽

The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

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