A BOUND FOR THE CHROMATIC NUMBER OF (, GEM)-FREE GRAPHS

2019 ◽  
Vol 100 (2) ◽  
pp. 182-188
Author(s):  
KATHIE CAMERON ◽  
SHENWEI HUANG ◽  
OWEN MERKEL
Keyword(s):  
Chromatic Number ◽  
Maximum Size ◽  
Free Graph ◽  
Induced Subgraph ◽  
Subgraph Isomorphic

As usual, $P_{n}$ ($n\geq 1$) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$. A graph is called ($P_{5}$, gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$, $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$. We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ($P_{5}$, gem)-free graph $G$.

Total coloring of quasi-line graphs and inflated graphs

2020 ◽  
pp. 2150060
Author(s):  
S. Mohan ◽  
J. Geetha ◽  
K. Somasundaram
Keyword(s):  
Chromatic Number ◽  
Total Coloring ◽  
Line Graphs ◽  
Tight Bound ◽  
Free Graph ◽  
Induced Subgraph ◽  
Total Chromatic Number

A total coloring of a graph is an assignment of colors to all the elements (vertices and edges) of the graph such that no two adjacent or incident elements receive the same color. A claw-free graph is a graph that does not have [Formula: see text] as an induced subgraph. Quasi-line and inflated graphs are two well-known classes of claw-free graphs. In this paper, we prove that the quasi-line and inflated graphs are totally colorable. In particular, we prove the tight bound of the total chromatic number of some classes of quasi-line graphs and inflated graphs.

scholarly journals Bipartite Induced Density in Triangle-Free Graphs

10.37236/8650 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Rémi De Joannis de Verclos ◽  
Ross J. Kang ◽  
François Pirot
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Free Graph ◽  
Induced Subgraph ◽  
Logarithmic Factor ◽  
Fractional Chromatic Number ◽  
List Chromatic Number

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any  triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

scholarly journals Strong Cliques in Claw-Free Graphs

2021 ◽  
Author(s):  
Michał Dębski ◽  
Małgorzata Śleszyńska-Nowak
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Maximum Size ◽  
Chromatic Index ◽  
Free Graph ◽  
Famous Conjecture ◽  
Strong Chromatic Index

AbstractFor a graph G, $$L(G)^2$$ L ( G ) 2 is the square of the line graph of G – that is, vertices of $$L(G)^2$$ L ( G ) 2 are edges of G and two edges $$e,f\in E(G)$$ e , f ∈ E ( G ) are adjacent in $$L(G)^2$$ L ( G ) 2 if at least one vertex of e is adjacent to a vertex of f and $$e\ne f$$ e ≠ f . The strong chromatic index of G, denoted by $$s'(G)$$ s ′ ( G ) , is the chromatic number of $$L(G)^2$$ L ( G ) 2 . A strong clique in G is a clique in $$L(G)^2$$ L ( G ) 2 . Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree $$\varDelta $$ Δ is at most $$\varDelta ^2 + \frac{1}{2}\varDelta $$ Δ 2 + 1 2 Δ . This result improves the only known result $$1.125\varDelta ^2+\varDelta $$ 1.125 Δ 2 + Δ , which is a bound for the strong chromatic index of claw-free graphs.

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 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.

Inductive graph invariants and approximation algorithms

2021 ◽  
Author(s):  
C. R. Subramanian
Keyword(s):  
Optimization Problems ◽  
Independence Number ◽  
Optimal Solution ◽  
Maximum Size ◽  
Chordal Graphs ◽  
Minimum Size ◽  
Optimal Size ◽  
Induced Subgraph ◽  
Special Cases ◽  
Optimal Formula

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

scholarly journals Total Domination and Matching Numbers in Claw-Free Graphs

10.37236/1085 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Matching Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Free Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $M$ of edges of a graph $G$ is a matching if no two edges in $M$ are incident to the same vertex. The matching number of $G$ is the maximum cardinality of a matching of $G$. A set $S$ of vertices in $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. If $G$ does not contain $K_{1,3}$ as an induced subgraph, then $G$ is said to be claw-free. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number. In this paper, we use transversals in hypergraphs to characterize connected claw-free graphs with minimum degree at least three that have equal total domination and matching numbers.

scholarly journals Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths

10.37236/6768 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Recent Result ◽  
Chromatic Number ◽  
Clique Number ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Large Chromatic Number ◽  
Vertex Colouring

We prove that for all integers $\kappa, s\ge 0$ there exists $c$ with the following property. Let $G$ be a graph with clique number at most $\kappa$ and chromatic number more than $c$. Then for every vertex-colouring (not necessarily optimal) of $G$, some induced subgraph of $G$ is an $s$-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016) who proved the same for graphs $G$ with $\kappa=2$ and girth at least five.

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.

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