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 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 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 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 $\Omega(\log_{h(a,t)} n)$.

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 Squared Chromatic Number Without Claws or Large Cliques

2019 ◽  
Vol 62 (1) ◽  
pp. 23-35
Author(s):  
Wouter Cames van Batenburg ◽  
Ross J. Kang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Clique Number ◽  
Sharp Bound ◽  
Line Graphs ◽  
Free Graph ◽  
Simple Graphs ◽  
Strengthened Form

AbstractLet $G$ be a claw-free graph on $n$ vertices with clique number $\unicode[STIX]{x1D714}$, and consider the chromatic number $\unicode[STIX]{x1D712}(G^{2})$ of the square $G^{2}$ of $G$. Writing $\unicode[STIX]{x1D712}_{s}^{\prime }(d)$ for the supremum of $\unicode[STIX]{x1D712}(L^{2})$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\unicode[STIX]{x1D712}(G^{2})\leqslant \unicode[STIX]{x1D712}_{s}^{\prime }(\unicode[STIX]{x1D714})$ for $\unicode[STIX]{x1D714}\in \{3,4\}$. For $\unicode[STIX]{x1D714}=3$, this implies the sharp bound $\unicode[STIX]{x1D712}(G^{2})\leqslant 10$. For $\unicode[STIX]{x1D714}=4$, this implies $\unicode[STIX]{x1D712}(G^{2})\leqslant 22$, which is within 2 of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erdős and Nešetřil.

scholarly journals On Some Interesting Ternary Formulas

10.37236/7901 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Pascal Ochem ◽  
Matthieu Rosenfeld
Keyword(s):  
Fixed Point ◽  
Maximum Degree ◽  
Minor Free Graphs

We obtain the following results about the avoidance of ternary formulas. Up to renaming of the letters, the only infinite ternary words avoiding the formula $ABCAB.ABCBA.ACB.BAC$ (resp. $ABCA.BCAB.BCB.CBA$) have the same set of recurrent factors as the fixed point of $0\mapsto 012$, $1\mapsto 02$, $2\mapsto 1$. The formula $ABAC.BACA.ABCA$ is avoided by polynomially many binary words and there exists arbitrarily many infinite binary words with different sets of recurrent factors that avoid it. If every variable of a ternary formula appears at least twice in the same fragment, then the formula is $3$-avoidable. The pattern $ABACADABCA$ is unavoidable for the class of $C_4$-minor-free graphs with maximum degree~$3$. This disproves a conjecture of Grytczuk. The formula $ABCA.ACBA$, or equivalently the palindromic pattern $ABCADACBA$, has avoidability index $4$.

scholarly journals The Degree-Diameter Problem for Sparse Graph Classes

10.37236/4313 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Guillermo Pineda-Villavicencio ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Classes ◽  
Minor Free Graphs

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.

A note on the strong edge-coloring of outerplanar graphs with maximum degree 3

2016 ◽  
Vol 32 (4) ◽  
pp. 883-890
Author(s):  
Shun-yi Liu ◽  
He-ping Zhang ◽  
Hong-liang Lu ◽  
Yu-qing Lin
Keyword(s):  
Maximum Degree ◽  
Edge Coloring ◽  
Outerplanar Graphs

scholarly journals Upper bounds on Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

2017 ◽  
Vol 32 ◽  
pp. 447-453
Author(s):  
Qi Kong ◽  
Ligong Wang
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Strongly Regular Graph ◽  
Upper Bounds ◽  
Free Graph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Strongly Regular

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. The results are the following: (1) Let $1< k\leq l< \Delta < n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3\Delta+k-2l+1+\sqrt{(3\Delta+k-2l+1)^{2}+16l(\Delta+n-1)}$$ with equality if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$

scholarly journals Repetition Number of Graphs

10.37236/96 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yair Caro ◽  
Douglas B. West
Keyword(s):  
Minimum Degree ◽  
Line Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Maximal Outerplanar Graphs ◽  
Vertex Graph ◽  
Repetition Number ◽  
Maximum Multiplicity

Every $n$-vertex graph has two vertices with the same degree (if $n\ge2$). In general, let rep$(G)$ be the maximum multiplicity of a vertex degree in $G$. An easy counting argument yields rep$(G)\ge n/(2d-2s+1)$, where $d$ is the average degree and $s$ is the minimum degree of $G$. Equality can hold when $2d$ is an integer, and the bound is approximately sharp in general, even when $G$ is restricted to be a tree, maximal outerplanar graph, planar triangulation, or claw-free graph. Among large claw-free graphs, repetition number $2$ is achievable, but if $G$ is an $n$-vertex line graph, then rep$(G)\ge{1\over4}n^{1/3}$. Among line graphs of trees, the minimum repetition number is $\Theta(n^{1/2})$. For line graphs of maximal outerplanar graphs, trees with perfect matchings, or triangulations with 2-factors, the lower bound is linear.

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