7 research outputs found
Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and
how excluding one fixed cycle length affects the number of colours
required as . For vertex-colouring and , if any two
distinct vertices connected by a path of at most edges are required to be
coloured differently, then a reduction by a logarithmic (in ) factor against
the trivial bound can be obtained by excluding an odd cycle length
if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by
a path of fewer than edges are required to be coloured differently, then
excluding an even cycle length is sufficient for a logarithmic
factor reduction. For , neither of the above statements are possible
for other parity combinations of and . These results can be
considered extensions of results due to Johansson (1996) and Mahdian (2000),
and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang
(2014).Comment: 14 pages, 1 figur
Optimization of eigenvalue bounds for the independence and chromatic number of graph powers
© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The k-thpower of a graph G=(V,E), G^k, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of G^k which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.The research of A. Abiad is partially supported by the FWO grant 1285921N. A. Abiad and M.A. Fiol gratefully acknowledge the support from DIAMANT. This research of M.A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. B. Nogueira acknowledges grant PRPQ/ADRC from UFMG. The authors would also like to thank Anurag Bishnoi for noticing a tight family for our bound (19).Peer ReviewedPostprint (author's final draft
Strong cliques and forbidden cycles
Given a graph , the strong clique number of is the
cardinality of a largest collection of edges every pair of which are incident
or connected by an edge in . We study the strong clique number of graphs
missing some set of cycle lengths. For a graph of large enough maximum
degree , we show among other results the following:
if is triangle-free;
if is -free; if
is -free for some . These bounds are attained by natural
extremal examples. Our work extends and improves upon previous work of Faudree,
Gy\'arf\'as, Schelp and Tuza (1990), Mahdian (2000) and Faron and Postle
(2019). We are motivated by the corresponding problems for the strong chromatic
index.Comment: 24 pages, 2 figure
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric