Locality in distributed graph algorithms
SIAM Journal on Computing
List edge chromatic number of graphs with large girth
Discrete Mathematics - Special volume (part two) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs” (“The theory of regular graphs”)
Applying randomized edge coloring algorithms to distributed communication: an experimental study
Proceedings of the seventh annual ACM symposium on Parallel algorithms and architectures
On the complexity of distributed network decomposition
Journal of Algorithms
Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds
SIAM Journal on Computing
Nearly optimal distributed edge coloring in O(log log n) rounds
Random Structures & Algorithms
List edge and list total colourings of multigraphs
Journal of Combinatorial Theory Series B
Near-optimal, distributed edge colouring via the nibble method
ESA '95 Selected papers from the third European symposium on Algorithms
Asymptotics of the list-chromatic index for multigraphs
Random Structures & Algorithms
Distributed O(Delta log(n))-Edge-Coloring Algorithm
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
List Total Colourings of Graphs
Combinatorics, Probability and Computing
End-to-end packet-scheduling in wireless ad-hoc networks
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Total Colorings Of Degenerate Graphs
Combinatorica
The Local Nature of List Colorings for Graphs of High Girth
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
| Hi-index | 0.00 |
We consider list coloring problems for graphs $\mathcal{G}$ of girth larger than $c\log_{\Delta-1}n$, where $n$ and $\Delta\geq3$ are, respectively, the order and the maximum degree of $\mathcal{G}$, and $c$ is a suitable constant. First, we determine that the edge and total list chromatic numbers of these graphs are $\chi'_l(\mathcal{G})=\Delta$ and $\chi”_l(\mathcal{G})=\Delta+1$. This proves that the general conjectures of Bollobás and Harris [Graphs Combin., 1 (1985), pp. 115-127], Behzad [The total chromatic number, in Combinatorial Mathematics and Its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, pp. 1-8], Vizing [Diskret. Analiz., 3 (1964), pp. 25-30], and Juvan, Mohar, and Škrekovski [Combin. Probab. Comput., 7 (1998), pp. 181-188] hold for this particular class of graphs. Moreover, our proofs exhibit a certain degree of “locality,” which we exploit to obtain an efficient distributed algorithm able to compute both kinds of optimal list colorings. Also, using an argument similar to one of Erdös, we show that our algorithm can compute $k$-list vertex colorings of graphs having girth larger than $c\log_{k-1}n$.