Clustered 3-colouring graphs of bounded degree

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

Partitioning Sparse Graphs into an Independent Set and a Forest of Bounded Degree

10.37236/6815 ◽

2018 ◽

Vol 25 (1) ◽

Author(s):

François Dross ◽

Mickael Montassier ◽

Alexandre Pinlou

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Independent Set ◽

Average Degree ◽

Bounded Degree ◽

Sparse Graphs ◽

Maximum Average Degree

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition. It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).

The Harmonious Chromatic Number of Bounded Degree Trees

Combinatorics Probability Computing ◽

10.1017/s0963548300001802 ◽

1996 ◽

Vol 5 (1) ◽

pp. 15-28 ◽

Cited By ~ 5

Author(s):

Keith Edwards

Keyword(s):

Positive Integer ◽

Chromatic Number ◽

Maximum Degree ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Simple Graph ◽

Bounded Degree ◽

Bounded Degree Trees ◽

Fixed Positive Integer ◽

Proper Vertex

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

Combinatorics Probability Computing ◽

10.1017/s0963548307008619 ◽

2008 ◽

Vol 17 (2) ◽

pp. 265-270 ◽

Cited By ~ 53

Author(s):

H. A. KIERSTEAD ◽

A. V. KOSTOCHKA

Keyword(s):

Positive Integer ◽

Polynomial Time ◽

Maximum Degree ◽

Short Proof ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Proper Vertex ◽

Colour Classes ◽

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

Locally Identifying Coloring of Graphs

10.37236/2417 ◽

2012 ◽

Vol 19 (2) ◽

Cited By ~ 6

Author(s):

Louis Esperet ◽

Sylvain Gravier ◽

Mickaël Montassier ◽

Pascal Ochem ◽

Aline Parreau

Keyword(s):

Bipartite Graph ◽

Planar Graphs ◽

Maximum Degree ◽

Perfect Graphs ◽

Vertex Coloring ◽

Complete Problem ◽

Minimum Number ◽

Np Complete ◽

Large Girth ◽

Proper Vertex

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring $c$ of a graph $G$ is said to be locally identifying, if for any adjacent vertices $u$ and $v$ with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of $u$ and $v$, respectively, are distinct. Let $\chi_{\rm{lid}}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of $G$. In this paper, we give several bounds on $\chi_{\rm{lid}}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{\rm{lid}}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.

Linear choosability of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3434 ◽

2005 ◽

Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽

Author(s):

Louis Esperet ◽

Mickael Montassier ◽

André Raspaud

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Oriented Graph ◽

Vertex Coloring ◽

Proper Coloring ◽

Maximum Average Degree ◽

Complete Problem ◽

International Audience ◽

Np Complete ◽

Proper Vertex

International audience A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete problem.

On Harmonious Colouring of Trees

10.37236/9 ◽

2012 ◽

Vol 19 (1) ◽

Cited By ~ 4

Author(s):

A. Aflaki ◽

S. Akbari ◽

K.J. Edwards ◽

D.S. Eskandani ◽

M. Jamaali ◽

...

Keyword(s):

Chromatic Number ◽

Maximum Degree ◽

Simple Graph ◽

Delta G ◽

Proper Vertex ◽

Let $G$ be a simple graph and $\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\Delta(T)\geq\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\Delta(T) \le \Big\lceil\frac{n}{2}\Big\rceil$, then there exists a harmonious colouring of $T$ with $\Big\lceil \frac{n}{2}\Big \rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\leq \Big\lceil \frac{n}{2} \Big\rceil +1$.

Tree-Like Distance Colouring for Planar Graphs of Sufficient Girth

10.37236/8220 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Ross J. Kang ◽

Willem Van Loon

Keyword(s):

Positive Integer ◽

Planar Graphs ◽

Maximum Degree ◽

Chromatic Index ◽

Constant Multiple ◽

Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges must receive different colours. Let $\pi'_t(d)$ and $\tau'_t(d)$ be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree $d$. We have that $\pi'_t(d)$ is at most and at least a non-trivial constant multiple larger than $\tau'_t(d)$. (We conjecture $\limsup_{d\to\infty}\pi'_2(d)/\tau'_2(d) =9/4$ in particular.) We prove for odd $t$ the existence of a quantity $g$ depending only on $t$ such that the distance-$t$ chromatic index of any planar multigraph of maximum degree $d$ and girth at least $g$ is at most $\tau'_t(d)$ if $d$ is sufficiently large. Such a quantity does not exist for even $t$. We also show a related, similar phenomenon for distance vertex-colouring.

The Complexity of Approximating the Matching Polynomial in the Complex Plane

ACM Transactions on Computation Theory ◽

10.1145/3448645 ◽

2021 ◽

Vol 13 (2) ◽

pp. 1-37

Author(s):

Ivona Bezáková ◽

Andreas Galanis ◽

Leslie Ann Goldberg ◽

Daniel Štefankovič

Keyword(s):

Real Axis ◽

Complex Plane ◽

Maximum Degree ◽

Negative Real Axis ◽

Positive Real ◽

Bounded Degree ◽

Matching Polynomial ◽

Correlation Decay ◽

Connective Constant ◽

General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.