scholarly journals Nonempty Intersection of Longest Paths in $2K_2$-Free Graphs

10.37236/7487 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Gili Golan ◽  
Songling Shan
Keyword(s):  
Maximum Degree ◽  
Short Note ◽  
Nonempty Intersection ◽  
Common Vertex ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Longest Paths ◽  
Split Graphs ◽  
Circular Arc Graphs

In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. In this short note, we show that, in nonempty $2K_2$-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty $2K_2$-free graph have a nonempty intersection. In particular, it strengthens the result on split graphs, as split graphs are $2K_2$-free.

scholarly journals Edge and Total Choosability of Near-Outerplanar Graphs

10.37236/1124 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Timothy J. Hetherington ◽  
Douglas R. Woodall
Keyword(s):  
Maximum Degree ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Minor Free Graphs ◽  
Straightforward Consequence ◽  
Edge Colouring

It is proved that, if $G$ is a $K_4$-minor-free graph with maximum degree $\Delta \ge 4$, then $G$ is totally $(\Delta+1)$-choosable; that is, if every element (vertex or edge) of $G$ is assigned a list of $\Delta+1$ colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ${\rm ch}"(G) = \chi"(G)$ for every graph $G$, is true for all $K_4$-minor-free graphs. The List-Edge-Colouring Conjecture is also known to be true for these graphs. As a fairly straightforward consequence, it is proved that both conjectures hold also for all $K_{2,3}$-minor free graphs and all $(\bar K_2 + (K_1 \cup K_2))$-minor-free graphs.

scholarly journals Total 4-Choosability of Series-Parallel Graphs

10.37236/1123 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Douglas R. Woodall
Keyword(s):  
Maximum Degree ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Minor Free Graphs

It is proved that, if $G$ is a $K_4$-minor-free graph with maximum degree 3, then $G$ is totally 4-choosable; that is, if every element (vertex or edge) of $G$ is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ${\rm ch}"(G) = \chi"(G)$ for every graph $G$, is true for all $K_4$-minor-free graphs and, therefore, for all outerplanar graphs.

scholarly journals WHEN DO THREE LONGEST PATHS HAVE A COMMON VERTEX?

2009 ◽  
Vol 01 (01) ◽  
pp. 115-120 ◽  
Author(s):  
MARIA AXENOVICH
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Common Vertex ◽  
Outerplanar Graphs ◽  
Connected Graphs ◽  
Longest Paths

It is well known that any two longest paths in a connected graph share a vertex. It is also known that there are connected graphs where 7 longest paths do not share a common vertex. It was conjectured that any three longest paths in a connected graph have a vertex in common. In this note we prove the conjecture for outerplanar graphs and give sufficient conditions for the conjecture to hold in general.

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

scholarly journals An efficientg-centroid location algorithm for cographs

2005 ◽  
Vol 2005 (9) ◽  
pp. 1405-1413 ◽  
Author(s):  
V. Prakash
Keyword(s):  
Location Problem ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Outerplanar Graphs ◽  
Arbitrary Graph ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Location Algorithm ◽  
Centroid Location

In 1998, Pandu Rangan et al. Proved that locating theg-centroid for an arbitrary graph is𝒩𝒫-hard by reducing the problem of finding the maximum clique size of a graph to theg-centroid location problem. They have also given an efficient polynomial time algorithm for locating theg-centroid for maximal outerplanar graphs, Ptolemaic graphs, and split graphs. In this paper, we present anO(nm)time algorithm for locating theg-centroid for cographs, wherenis the number of vertices andmis the number of edges of the graph.

Split Graphs with Specified Dilworth Numbers

1984 ◽  
Vol 27 (1) ◽  
pp. 43-47
Author(s):  
Chiê Nara ◽  
Iwao Sato
Keyword(s):  
Induced Subgraph ◽  
Split Graph ◽  
Complete Part ◽  
Split Graphs

AbstractLet G be a split graph with the independent part IG and the complete part KG. Suppose that the Dilworth number of (IG, ≼) with respect to the vicinal preorder ≼ is two and that of (KG, ≼) is an integer k. We show that G has a specified graph Hk, defined in this paper, as an induced subgraph.

scholarly journals Some New Classes of Open Distance-Pattern Uniform Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bibin K. Jose
Keyword(s):  
Nonempty Subset ◽  
Chordal Graphs ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Induced Subgraphs ◽  
Minimum Cardinality ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Strongly Chordal Graphs

Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph G can be embedded into a self-complementary odpu-graph H, such that G and G¯ are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.

scholarly journals The Chromatic Index of Proper Circular-arc Graphs of Odd Maximum Degree which are Chordal

2019 ◽  
Vol 346 ◽  
pp. 125-133
Author(s):  
João Pedro W. Bernardi ◽  
Murilo V.G. da Silva ◽  
André Luiz P. Guedes ◽  
Leandro M. Zatesko
Keyword(s):  
Maximum Degree ◽  
Chromatic Index ◽  
Circular Arc ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

scholarly journals Optimal induced universal graphs for bounded-degree graphs

2017 ◽  
Vol 166 (1) ◽  
pp. 61-74 ◽  
Author(s):  
NOGA ALON ◽  
RAJKO NENADOV
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Vertex Graph ◽  
Bounded Degree Graphs ◽  
Universal Graphs

AbstractWe show that for any constant Δ ≥ 2, there exists a graph Γ withO(nΔ / 2) vertices which contains everyn-vertex graph with maximum degree Δ as an induced subgraph. For odd Δ this significantly improves the best-known earlier bound and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(nΔ/2) vertices.

Trees with given maximum degree minimizing the spectral radius

2016 ◽  
Vol 31 ◽  
pp. 335-361
Author(s):  
Xue Du ◽  
Lingsheng Shi
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Equal Length ◽  
Disjoint Paths ◽  
Common Vertex ◽  
Largest Eigenvalue ◽  
Large N ◽  
The Largest Eigenvalue

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

Sign in / Sign up

Export Citation Format

Share Document