Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

ISRN Combinatorics ◽

10.1155/2013/673971 ◽

2013 ◽

Vol 2013 ◽

pp. 1-4

Author(s):

Zh. G. Nikoghosyan

Keyword(s):

Complete Graphs ◽

Connected Subgraph ◽

Forbidden Subgraph ◽

Hamilton Cycles ◽

Forbidden Subgraphs ◽

Free Graph ◽

Hamiltonian Graphs ◽

Connected Graphs ◽

Connected Subgraphs

In 1974, Goodman and Hedetniemi proved that every 2-connected -free graph is hamiltonian. This result gave rise many other conditions for Hamilton cycles concerning various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In this paper we investigate analogous problems when forbidden subgraphs are disconnected which affects more global structures in graphs such as tough structures instead of traditional connectivity structures. In 1997, it was proved that a single forbidden connected subgraph in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with . In this paper we prove that a single forbidden subgraph can create a non trivial class of hamiltonian graphs if is disconnected: every -free graph either is hamiltonian or belongs to a well defined class of non hamiltonian graphs; every 1-tough -free graph is hamiltonian. We conjecture that every 1-tough -free graph is hamiltonian and every 1-tough -free graph is hamiltonian.

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Toughness, Forbidden Subgraphs and Pancyclicity

Graphs and Combinatorics ◽

10.1007/s00373-021-02284-y ◽

2021 ◽

Vol 37 (3) ◽

pp. 839-866

Author(s):

Wei Zheng ◽

Hajo Broersma ◽

Ligong Wang

Keyword(s):

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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A Pair of Forbidden Subgraphs and 2-Factors

Combinatorics Probability Computing ◽

10.1017/s0963548311000514 ◽

2012 ◽

Vol 21 (1-2) ◽

pp. 149-158 ◽

Cited By ~ 2

Author(s):

JUN FUJISAWA ◽

AKIRA SAITO

Keyword(s):

Finite Number ◽

Connected Graph ◽

Minimum Degree ◽

Forbidden Subgraphs ◽

Free Graph ◽

Connected Graphs ◽

Sharp Difference ◽

Large Order

In this paper, we consider pairs of forbidden subgraphs that imply the existence of a 2-factor in a graph. For d ≥ 2, let d be the set of connected graphs of minimum degree at least d. Let F1 and F2 be connected graphs and let be a set of connected graphs. Then {F1, F2} is said to be a forbidden pair for if every {F1, F2}-free graph in of sufficiently large order has a 2-factor. Faudree, Faudree and Ryjáček have characterized all the forbidden pairs for the set of 2-connected graphs. We first characterize the forbidden pairs for 2, which is a larger set than the set of 2-connected graphs, and observe a sharp difference between the characterized pairs and those obtained by Faudree, Faudree and Ryjáček. We then consider the forbidden pairs for connected graphs of large minimum degree. We prove that if {F1, F2} is a forbidden pair for d, then either F1 or F2 is a star of order at most d + 2. Ota and Tokuda have proved that every $K_{1, \lfloor\frac{d+2}{2}\rfloor}$-free graph of minimum degree at least d has a 2-factor. These results imply that for k ≥ d + 2, no connected graphs F except for stars of order at most d + 2 make {K1,k, F} a forbidden pair for d, while for $k\le \bigl\lfloor\frac{d+2}{2} \bigr\rfloor$ every connected graph F makes {K1,k, F} a forbidden pair for d. We consider the remaining range of $\bigl\lfloor\frac{d+2}{2} \bigr\rfloor < k < d+2$, and prove that only a finite number of connected graphs F make {K1,k, F} a forbidden pair for d.

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Note on Highly Connected Monochromatic Subgraphs in $2$-Colored Complete Graphs

The Electronic Journal of Combinatorics ◽

10.37236/502 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 4

Author(s):

Shinya Fujita ◽

Colton Magnant

Keyword(s):

Complete Graph ◽

Complete Graphs ◽

Connected Subgraph ◽

Connected Subgraphs

In this note, we improve upon some recent results concerning the existence of large monochromatic, highly connected subgraphs in a $2$-coloring of a complete graph. In particular, we show that if $n\ge 6.5(k - 1)$, then in any $2$-coloring of the edges of $K_{n}$, there exists a monochromatic $k$-connected subgraph of order at least $n - 2(k - 1)$. Our result improves upon several recent results by a variety of authors.

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Structure and Substructure Connectivity of Hypercube-Like Networks

Parallel Processing Letters ◽

10.1142/s0129626420400071 ◽

2020 ◽

Vol 30 (03) ◽

pp. 2040007

Author(s):

Cheng-Kuan Lin ◽

Eddie Cheng ◽

László Lipták

Keyword(s):

Text Structure ◽

Network Architectures ◽

Connected Subgraph ◽

Menger's Theorem ◽

The Family ◽

Minimum Number ◽

Structure Formula ◽

Connected Subgraphs ◽

Twisted Cubes

The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] can be determined using Menger’s theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.

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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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