scholarly journals Defective Choosability of Graphs without Small Minors

10.37236/181 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rupert G. Wood ◽  
Douglas R. Woodall
Keyword(s):  
Triangular Prism ◽  
Free Graph ◽  
Minor Free Graphs ◽  
Proper Subgraph

For each proper subgraph $H$ of $K_5$, we determine all pairs $(k,d)$ such that every $H$-minor-free graph is $(k,d)^*$-choosable or $(k,d)^-$-choosable. The main structural lemma is that the only 3-connected $(K_5-e)$-minor-free graphs are wheels, the triangular prism, and $K_{3,3}$; this is used to prove that every $(K_5-e)$-minor-free graph is 4-choosable and $(3,1)$-choosable.

scholarly journals Density of universal classes of series-parallel graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Jaroslav Nešetřil ◽  
Yared Nigussie
Keyword(s):  
Partial Order ◽  
Chromatic Number ◽  
Rational Number ◽  
Free Graph ◽  
Circular Chromatic Number ◽  
Density Result ◽  
Proper Subclass ◽  
International Audience ◽  
Universal Classes ◽  
Minor Free Graphs

International audience A class of graphs $\mathcal{C}$ ordered by the homomorphism relation is universal if every countable partial order can be embedded in $\mathcal{C}$. It was shown in [ZH] that the class $\mathcal{C_k}$ of $k$-colorable graphs, for any fixed $k≥3$, induces a universal partial order. In [HN1], a surprisingly small subclass of $\mathcal{C_3}$ which is a proper subclass of $K_4$-minor-free graphs $(\mathcal{G/K_4)}$ is shown to be universal. In another direction, a density result was given in [PZ], that for each rational number $a/b ∈[2,8/3]∪ \{3\}$, there is a $K_4$-minor-free graph with circular chromatic number equal to $a/b$. In this note we show for each rational number $a/b$ within this interval the class $\mathcal{K_{a/b}}$ of $0K_4$-minor-free graphs with circular chromatic number $a/b$ is universal if and only if $a/b ≠2$, $5/2$ or $3$. This shows yet another surprising richness of the $K_4$-minor-free class that it contains universal classes as dense as the rational numbers.

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 Toughness of $K_{a,t}$-Minor-free Graphs

10.37236/635 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guantao Chen ◽  
Yoshimi Egawa ◽  
Ken-ichi Kawarabayashi ◽  
Bojan Mohar ◽  
Katsuhiro Ota
Keyword(s):  
Planar Graph ◽  
Positive Constant ◽  
Complete Graph ◽  
Planar Graphs ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Free Graph ◽  
Minor Free Graphs ◽  
Vertex Sets ◽  
Connected Planar Graph

The toughness of a non-complete graph $G$ is the minimum value of $\frac{|S|}{\omega(G-S)}$ among all separating vertex sets $S\subset V(G)$, where $\omega(G-S)\ge 2$ is the number of components of $G-S$. It is well-known that every $3$-connected planar graph has toughness greater than $1/2$. Related to this property, every $3$-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a $2$-walk, etc. Realizing that 3-connected planar graphs are essentially the same as 3-connected $K_{3,3}$-minor-free graphs, we consider a generalization to $a$-connected $K_{a,t}$-minor-free graphs, where $3\le a\le t$. We prove that there exists a positive constant $h(a,t)$ such that every $a$-connected $K_{a,t}$-minor-free graph $G$ has toughness at least $h(a,t)$. For the case where $a=3$ and $t$ is odd, we obtain the best possible value for $h(3,t)$. As a corollary it is proved that every such graph of order $n$ contains a cycle of length $\Omega(\log_{h(a,t)} n)$.

The Field Uniformity Analysis in a Triangular Prism Reverberation Chamber with a QRD

2011 ◽  
Vol E94-B (1) ◽  
pp. 334-337 ◽  
Author(s):  
Jung-Hoon KIM ◽  
Hye-Kwang KIM ◽  
Eugene RHEE ◽  
Sung-Il YANG
Keyword(s):  
Triangular Prism ◽  
Reverberation Chamber ◽  
Field Uniformity

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 On multiplicative degree based topological indices for planar octahedron networks

2020 ◽  
Vol 43 (1) ◽  
pp. 219-228
Author(s):  
Ghulam Dustigeer ◽  
Haidar Ali ◽  
Muhammad Imran Khan ◽  
Yu-Ming Chu
Keyword(s):  
Graph Theory ◽  
Information Science ◽  
Biological Activities ◽  
Molecular Graph ◽  
Triangular Prism ◽  
Structure Property ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Chemical Graph Theory ◽  
Analytical Formulas

AbstractChemical graph theory is a branch of graph theory in which a chemical compound is presented with a simple graph called a molecular graph. There are atomic bonds in the chemistry of the chemical atomic graph and edges. The graph is connected when there is at least one connection between its vertices. The number that describes the topology of the graph is called the topological index. Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. We evaluated the second multiplicative Zagreb index, first and second universal Zagreb indices, first and second hyper Zagreb indices, sum and product connectivity indices for the planar octahedron network, triangular prism network, hex planar octahedron network, and give these indices closed analytical formulas.

scholarly journals Universal and unavoidable graphs

2021 ◽  
pp. 1-14
Author(s):  
Matija Bucić ◽  
Nemanja Draganić ◽  
Benny Sudakov
Keyword(s):  
Free Graph ◽  
Turán Number ◽  
Work Done

Abstract The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

Sign in / Sign up

Export Citation Format

Share Document