Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth

1998 ◽  
Vol 7 (4) ◽  
pp. 375-386 ◽  
Author(s):  
THOMAS EMDEN-WEINERT ◽  
STEFAN HOUGARDY ◽  
BERND KREUTER
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Maximal Degree ◽  
Large Girth

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)[ges ]log[mid ]G[mid ]/13logk and maximum degree Δ(G)[les ]6k13 can exist. We also study several related problems.

scholarly journals On the complexity of the balanced vertex ordering problem

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Jan Kára ◽  
Jan Kratochvil ◽  
David R. Wood
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Graph Drawing ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Vertex Ordering ◽  
The Difference ◽  
Balanced Ordering ◽  
Applied Math

Graphs and Algorithms International audience We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.

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

2008 ◽  
Vol 17 (2) ◽  
pp. 265-270 ◽  
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 ◽  
Vertex Colouring

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.

scholarly journals Destroying Bicolored $P_3$s by Deleting Few Edges

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Niels Grüttemeier ◽  
Christian Komusiewicz ◽  
Jannik Schestag ◽  
Frank Sommer
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Red Edge ◽  
Bounded Degree Graphs

We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$. Comment: 25 pages

scholarly journals An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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