scholarly journals Spanning Trails in a 2-Connected Graph

10.37236/8121 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Shipeng Wang ◽  
Liming Xiong
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Free Graph ◽  
Delta G ◽  
Exceptional Families

In this article we prove the following: Let $G$ be a $2$-connected graph with circumference $c(G)$. If  $c(G)\leq 5$, then $G$ has a spanning trail starting from any vertex, if  $c(G)\leq 7$, then $G$ has a spanning trail.  As applications of  this result, we obtain the following. Every $2$-edge-connected graph of order at most 8 has a spanning trail starting from any vertex  with the exception of six graphs.  Let $G$ be a $2$-edge-connected graph and $S$ a subset of $V(G)$ such that $E(G-S)=\emptyset$ and $|S|\leq 6$. Then $G$ has a trail traversing all vertices of $S$ with the exception of two graphs, moreover, if $|S|\leq 4$, then $G$ has a trail starting from any vertex of $S$ and containing $S$. Every $2$-connected claw-free graph $G$ with order $n$ and minimum degree $\delta(G)> \frac{n}{7}+4\geq 23$ is traceable or belongs to two exceptional families of well-defined  graphs, and moreover, if $\delta(G)> \frac{n}{6}+4\geq 13$, then $G$ is traceable. All above results are sharp in a sense.

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.

Radius, Diameter and the Degree Sequence of a Graph

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Jaya Percival Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Free Graph

AbstractWe give asymptotically sharp upper bounds on the radius and diameter of(i) a connected graph,(ii) a connected triangle-free graph,(iii) a connected C4-free graph of given order, minimum degree, and maximum degree.We also give better bounds on the radius and diameter for triangle-free graphs with a given order, minimum degree and a given number of distinct terms in the degree sequence of the graph. Our results improve on old classical theorems by Erd˝os, Pach, Pollack and Tuza [Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47 (1989), 73-79] on radius, diameter and minimum degree.

A Pair of Forbidden Subgraphs and 2-Factors

2012 ◽  
Vol 21 (1-2) ◽  
pp. 149-158 ◽  
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.

EVERY N2-LOCALLY CONNECTED CLAW-FREE GRAPH WITH MINIMUM DEGREE AT LEAST 7 IS Z3-CONNECTED

2011 ◽  
Vol 03 (02) ◽  
pp. 193-201
Author(s):  
ERLING WEI ◽  
YE CHEN ◽  
PING LI ◽  
HONG-JIAN LAI
Keyword(s):  
Abelian Group ◽  
Connected Graph ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Free Graph

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A-{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b : V(G) ↦ A satisfying Συ∈V(G)b(υ) = 0, there is a function f : E(G) ↦ A* such that for each vertex υ ∈ V(G), the total amount of f values on the edges directed out from υ minus the total amount of f values on the edges directed into υ equals b(υ). Let Z3denote the group of order 3. Jaeger et al. conjectured that there exists an integer k such that every k-edge-connected graph is Z3-connected. In this paper, we prove that every N2-locally connected claw-free graph G with minimum degree δ(G) ≥ 7 is Z3-connected.

scholarly journals Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

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].

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

scholarly journals Thek-Core and Branching Processes

2008 ◽  
Vol 17 (1) ◽  
pp. 111-136 ◽  
Author(s):  
OLIVER RIORDAN
Keyword(s):  
Power Law ◽  
Branching Process ◽  
Minimum Degree ◽  
Branching Processes ◽  
Threshold Value ◽  
Free Graph ◽  
Scale Free ◽  
Asymptotic Size ◽  
Inhomogeneous Random Graphs ◽  
Local Coupling

Thek-coreof a graphGis the maximal subgraph ofGhaving minimum degree at leastk. In 1996, Pittel, Spencer and Wormald found the threshold λcfor the emergence of a non-trivialk-core in the random graphG(n, λ/n), and the asymptotic size of thek-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study thek-core in a certain power-law or ‘scale-free’ graph with a parameterccontrolling the overall density of edges. For eachk≥ 3, we find the threshold value ofcat which thek-core emerges, and the fraction of vertices in thek-core whencis ϵ above the threshold. In contrast toG(n, λ/n), this fraction tends to 0 as ϵ→0.

scholarly journals A NEIGHBOURHOOD CONDITION FOR GRAPHS TO BE FRACTIONAL (k, m)-DELETED GRAPHS

2009 ◽  
Vol 52 (1) ◽  
pp. 33-40 ◽  
Author(s):  
SIZHONG ZHOU
Keyword(s):  
Connected Graph ◽  
Delta G

AbstractLet G be a connected graph of order n, and let k ≥ 2 and m ≥ 0 be two integers. In this paper, we show that G is a fractional (k, m)-deleted graph if $\delta(G)\,{\geq}\, k+m+\frac{(m+1)^{2}-1}{4k}$, $n\,{\geq}\, 9k-1-4\sqrt{2(k-1)^{2}+2}+2(2k+1)m$ and $|N_G(x)\cup N_G(y)|\,{\geq}\,\frac{1}{2}(n+k-2)$ for each pair of non-adjacent vertices x, y of G. This result is an extension of the previous result of Zhou [11].

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

Sign in / Sign up

Export Citation Format

Share Document