scholarly journals A Degree Sum Condition on the Order, the Connectivity and the Independence Number for Hamiltonicity

10.37236/5480 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Shuya Chiba ◽  
Michitaka Furuya ◽  
Kenta Ozeki ◽  
Masao Tsugaki ◽  
Tomoki Yamashita
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Independence Number ◽  
Hamilton Cycle ◽  
Degree Sum ◽  
The Third ◽  
Degree Sum Condition

In [Graphs Combin. 24 (2008) 469–483], the third author and the fifth author conjectured that if $G$ is a $k$-connected graph such that $\sigma_{k+1}(G) \ge |V(G)|+\kappa(G)+(k-2)(\alpha(G)-1)$, then $G$ contains a Hamilton cycle, where $\sigma_{k+1}(G)$, $\kappa(G)$ and $\alpha(G)$ are the minimum degree sum of $k+1$ independent vertices, the connectivity and the independence number of $G$, respectively. In this paper, we settle this conjecture. The degree sum condition is best possible.  

scholarly journals Spanning $k$-trees of Bipartite Graphs

10.37236/3628 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mikio Kano ◽  
Kenta Ozeki ◽  
Kazuhiro Suzuki ◽  
Masao Tsugaki ◽  
Tomoki Yamashita
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Degree Sum

A tree is called a $k$-tree if its maximum degree is at most $k$. We prove the following theorem. Let $k \geq 2$ be an integer, and $G$ be a connected bipartite graph with bipartition $(A,B)$ such that $|A| \le |B| \le (k-1)|A|+1$. If $\sigma_k(G) \ge |B|$, then $G$ has a spanning $k$-tree, where $\sigma_k(G)$ denotes the minimum degree sum of $k$ independent vertices of $G$. Moreover, the condition on $\sigma_k(G)$ is sharp. It was shown by Win (Abh. Math. Sem. Univ. Hamburg, 43, 263–267, 1975) that if a connected graph $H$ satisfies $\sigma_k(H) \ge |H|-1$, then $H$ has a spanning $k$-tree. Thus our theorem shows that the condition becomes much weaker if the graph is bipartite.

On a Sharp Degree Sum Condition for Disjoint Chorded Cycles in Graphs

2010 ◽  
Vol 26 (2) ◽  
pp. 173-186 ◽  
Author(s):  
Shuya Chiba ◽  
Shinya Fujita ◽  
Yunshu Gao ◽  
Guojun Li
Keyword(s):  
Degree Sum ◽  
Chorded Cycles ◽  
Degree Sum Condition

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals Long Path Lemma concerning Connectivity and Independence Number

10.37236/636 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Shinya Fujita ◽  
Alexander Halperin ◽  
Colton Magnant
Keyword(s):  
Connected Graph ◽  
Independence Number

We show that, in a $k$-connected graph $G$ of order $n$ with $\alpha(G) = \alpha$, between any pair of vertices, there exists a path $P$ joining them with $$|P| \geq \min \left\{ n, \frac{(k - 1)(n - k)}{\alpha} + k \right\}.$$ This implies that, for any edge $e \in E(G)$, there is a cycle containing $e$ of length at least $$\min \left\{ n, \frac{(k - 1)(n - k)}{\alpha} + k \right\}.$$ Moreover, we generalize our result as follows: for any choice $S$ of $s \leq k$ vertices in $G$, there exists a tree $T$ whose set of leaves is $S$ with $$|T| \geq \min \left\{ n, \frac{(k - s + 1)(n - k)}{\alpha} + k \right\}.$$

scholarly journals On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

10.37236/5173 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Connected Graph ◽  
Special Class ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Approach ◽  
The Family ◽  
Edge Colourings ◽  
Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

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