A proof of a conjecture on the differential of a subcubic graph

2020 ◽  
Vol 287 ◽  
pp. 27-39
Author(s):  
R. Khoeilar ◽  
H. Karami ◽  
M. Chellali ◽  
S.M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Subcubic Graph

scholarly journals Large independent sets in triangle-free cubic graphs: beyond planarity

2020 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Gwenaël Joret ◽  
Jan Goedgebeur
Keyword(s):  
Planar Graph ◽  
Present Article ◽  
Additional Assumption ◽  
Independence Number ◽  
Independent Set ◽  
Cubic Graphs ◽  
Subcubic Graph ◽  
Color Class ◽  
Subcubic Graphs ◽  
Colorable Graph

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

scholarly journals Packing Three-Vertex Paths in a Subcubic Graph

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Adrian Kosowski ◽  
Michal Malafiejski ◽  
Pawel Zyliński
Keyword(s):  
Linear Time ◽  
Packing Problem ◽  
Time Algorithm ◽  
Vertex Degree ◽  
Linear Time Algorithm ◽  
Cubic Graph ◽  
Subcubic Graph ◽  
International Audience ◽  
Subcubic Graphs

International audience In our paper we consider the $P_3$-packing problem in subcubic graphs of different connectivity, improving earlier results of Kelmans and Mubayi. We show that there exists a $P_3$-packing of at least $\lceil 3n/4\rceil$ vertices in any connected subcubic graph of order $n>5$ and minimum vertex degree $\delta \geq 2$, and that this bound is tight. The proof is constructive and implied by a linear-time algorithm. We use this result to show that any $2$-connected cubic graph of order $n>8$ has a $P_3$-packing of at least $\lceil 7n/9 \rceil$ vertices.

On the hardness of finding the geodetic number of a subcubic graph

2018 ◽  
Vol 135 ◽  
pp. 22-27 ◽  
Author(s):  
Letícia R. Bueno ◽  
Lucia D. Penso ◽  
Fábio Protti ◽  
Victor R. Ramos ◽  
Dieter Rautenbach ◽  
...  
Keyword(s):  
Subcubic Graph ◽  
Geodetic Number

Coloring 3-power of 3-subdivision of subcubic graph

2018 ◽  
Vol 10 (03) ◽  
pp. 1850041 ◽  
Author(s):  
Fang Wang ◽  
Xiaoping Liu
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
London Math ◽  
Total Coloring ◽  
Total Graph ◽  
Total Chromatic Number ◽  
Subcubic Graph ◽  
Minimum Integer ◽  
Vertex Set ◽  
Number Formula

Let [Formula: see text] be a graph and [Formula: see text] be a positive integer. The [Formula: see text]-subdivision [Formula: see text] of [Formula: see text] is the graph obtained from [Formula: see text] by replacing each edge by a path of length [Formula: see text]. The [Formula: see text]-power [Formula: see text] of [Formula: see text] is the graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if the distance [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] is at most [Formula: see text]. Note that [Formula: see text] is the total graph [Formula: see text] of [Formula: see text]. The chromatic number [Formula: see text] of [Formula: see text] is the minimum integer [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-coloring. The total chromatic number of [Formula: see text], denoted by [Formula: see text], is the chromatic number of [Formula: see text]. Rosenfeld [On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396–402] and independently, Vijayaditya [On total chromatic number of a graph, J. London Math. Soc. 2 (1971) 405–408] showed that for a subcubic graph [Formula: see text], [Formula: see text]. In this note, we prove that for a subcubic graph [Formula: see text], [Formula: see text].

scholarly journals Extremal $H$-Free Planar Graphs

10.37236/8255 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Yongxin Lan ◽  
Yongtang Shi ◽  
Zi-Xia Song
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Sufficient Conditions ◽  
Subcubic Graph ◽  
Stone Theorem

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory  83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield  $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem.  We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to  $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

scholarly journals Strong List Edge Coloring of Subcubic Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Hongping Ma ◽  
Zhengke Miao ◽  
Hong Zhu ◽  
Jianhua Zhang ◽  
Rong Luo
Keyword(s):  
Edge Coloring ◽  
Average Degree ◽  
Maximum Average Degree ◽  
Subcubic Graph ◽  
List Edge Coloring ◽  
Subcubic Graphs

We study strong list edge coloring of subcubic graphs, and we prove that every subcubic graph with maximum average degree less than 15/7, 27/11, 13/5, and 36/13 can be strongly list edge colored with six, seven, eight, and nine colors, respectively.

scholarly journals Permanent Index of Matrices Associated with Graphs

10.37236/5494 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Tsai-Lien Wong ◽  
Xuding Zhu
Keyword(s):  
Graph Theory ◽  
Real Number ◽  
Reduction Method ◽  
Total Weight ◽  
Combinatorial Nullstellensatz ◽  
Induced Subgraphs ◽  
Halin Graph ◽  
Subcubic Graph ◽  
Promising Tool ◽  
Total Weighting

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $\phi_f(u) \ne \phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz.  This approach leads to conjectures on the permanent indices of matrices  $A_G$ and $B_G$ associated to a graph $G$. In this  paper, we establish a method that reduces the study of permanent of  matrices associated to a graph $G$ to the study of permanent of  matrices associated to induced subgraphs of $G$. Using this  reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable.

scholarly journals List-coloring the square of a subcubic graph

2007 ◽  
Vol 57 (1) ◽  
pp. 65-87 ◽  
Author(s):  
Daniel W. Cranston ◽  
Seog-Jin Kim
Keyword(s):  
List Coloring ◽  
Subcubic Graph

scholarly journals On Oriented Arc-Coloring of Subcubic Graphs

10.37236/1095 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Alexandre Pinlou
Keyword(s):  
Oriented Graph ◽  
Chromatic Index ◽  
Subcubic Graph ◽  
Line Digraph ◽  
Minimum Number ◽  
Subcubic Graphs

A homomorphism from an oriented graph $G$ to an oriented graph $H$ is a mapping $\varphi$ from the set of vertices of $G$ to the set of vertices of $H$ such that $\overrightarrow{\varphi(u)\varphi(v)}$ is an arc in $H$ whenever $\overrightarrow{uv}$ is an arc in $G$. The oriented chromatic index of an oriented graph $G$ is the minimum number of vertices in an oriented graph $H$ such that there exists a homomorphism from the line digraph $LD(G)$ of $G$ to $H$ (Recall that $LD(G)$ is given by $V(LD(G))=A(G)$ and $ \overrightarrow{ab}\in A(LD(G))$ whenever $a=\overrightarrow{uv}$ and $b=\overrightarrow{vw}$). We prove that every oriented subcubic graph has oriented chromatic index at most $7$ and construct a subcubic graph with oriented chromatic index $6$.

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