Matches in SemOpenAlex for { <https://semopenalex.org/work/W3203673768> ?p ?o ?g. }
Showing items 1 to 72 of
72
with 100 items per page.
- W3203673768 abstract "By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $Delta$ has chromatic number at most $(C+o(1))Delta/log Delta$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one can in fact take $C = 1$. Here we show that for every $q geq (1 + o(1))Delta/log Delta$, the number $c(G,q)$ of proper $q$-colorings of $G$ satisfies $c(G, q) ,geq, left(1 - frac{1}{q}right)^m ((1-o(1))q)^n$, where $n = |V(G)|$ and $m = |E(G)|$. Except for the $o(1)$ term, this lower bound is best possible as witnessed by random $Delta$-regular graphs. When $q = (1 + o(1)) Delta/log Delta$, our result yields the inequality $c(G,q) ,geq, expleft((1 - o(1)) frac{log Delta}{2} nright)$, which improves an earlier bound of Iliopoulos and yields the optimal value for the constant factor in the exponent. Furthermore, this result implies the optimal lower bound on the number of independent sets in $G$ due to Davies, Jenssen, Perkins, and Roberts. An important ingredient in our proof is the counting method that was recently developed by Rosenfeld. As a byproduct, we obtain an alternative proof of Molloy's bound $chi(G) leq (1 + o(1))Delta/log Delta$ using Rosenfeld's method in place of entropy compression (other proofs of Molloy's theorem using Rosenfeld's technique were given independently by Hurley and Pirot and Martinsson)." @default.
- W3203673768 created "2021-10-11" @default.
- W3203673768 creator A5019172041 @default.
- W3203673768 creator A5024996448 @default.
- W3203673768 creator A5042018306 @default.
- W3203673768 creator A5082937606 @default.
- W3203673768 date "2021-09-27" @default.
- W3203673768 modified "2023-09-23" @default.
- W3203673768 title "Counting colorings of triangle-free graphs" @default.
- W3203673768 cites W1592213313 @default.
- W3203673768 cites W2018483383 @default.
- W3203673768 cites W2066616484 @default.
- W3203673768 cites W2068871408 @default.
- W3203673768 cites W2091476183 @default.
- W3203673768 cites W2101851522 @default.
- W3203673768 cites W2329508558 @default.
- W3203673768 cites W2548129158 @default.
- W3203673768 cites W2963609862 @default.
- W3203673768 cites W2963927233 @default.
- W3203673768 cites W3014235966 @default.
- W3203673768 cites W3106164415 @default.
- W3203673768 doi "https://doi.org/10.48550/arxiv.2109.13376" @default.
- W3203673768 hasPublicationYear "2021" @default.
- W3203673768 type Work @default.
- W3203673768 sameAs 3203673768 @default.
- W3203673768 citedByCount "1" @default.
- W3203673768 countsByYear W32036737682021 @default.
- W3203673768 crossrefType "posted-content" @default.
- W3203673768 hasAuthorship W3203673768A5019172041 @default.
- W3203673768 hasAuthorship W3203673768A5024996448 @default.
- W3203673768 hasAuthorship W3203673768A5042018306 @default.
- W3203673768 hasAuthorship W3203673768A5082937606 @default.
- W3203673768 hasBestOaLocation W32036737681 @default.
- W3203673768 hasConcept C106301342 @default.
- W3203673768 hasConcept C114614502 @default.
- W3203673768 hasConcept C118615104 @default.
- W3203673768 hasConcept C121332964 @default.
- W3203673768 hasConcept C134306372 @default.
- W3203673768 hasConcept C138885662 @default.
- W3203673768 hasConcept C2780388253 @default.
- W3203673768 hasConcept C33923547 @default.
- W3203673768 hasConcept C41895202 @default.
- W3203673768 hasConcept C62520636 @default.
- W3203673768 hasConcept C77553402 @default.
- W3203673768 hasConceptScore W3203673768C106301342 @default.
- W3203673768 hasConceptScore W3203673768C114614502 @default.
- W3203673768 hasConceptScore W3203673768C118615104 @default.
- W3203673768 hasConceptScore W3203673768C121332964 @default.
- W3203673768 hasConceptScore W3203673768C134306372 @default.
- W3203673768 hasConceptScore W3203673768C138885662 @default.
- W3203673768 hasConceptScore W3203673768C2780388253 @default.
- W3203673768 hasConceptScore W3203673768C33923547 @default.
- W3203673768 hasConceptScore W3203673768C41895202 @default.
- W3203673768 hasConceptScore W3203673768C62520636 @default.
- W3203673768 hasConceptScore W3203673768C77553402 @default.
- W3203673768 hasLocation W32036737681 @default.
- W3203673768 hasOpenAccess W3203673768 @default.
- W3203673768 hasPrimaryLocation W32036737681 @default.
- W3203673768 hasRelatedWork W2029930003 @default.
- W3203673768 hasRelatedWork W2056549194 @default.
- W3203673768 hasRelatedWork W2064339860 @default.
- W3203673768 hasRelatedWork W2141385105 @default.
- W3203673768 hasRelatedWork W2378464885 @default.
- W3203673768 hasRelatedWork W2794380282 @default.
- W3203673768 hasRelatedWork W2952361517 @default.
- W3203673768 hasRelatedWork W3016091878 @default.
- W3203673768 hasRelatedWork W3103409729 @default.
- W3203673768 hasRelatedWork W4287017524 @default.
- W3203673768 isParatext "false" @default.
- W3203673768 isRetracted "false" @default.
- W3203673768 magId "3203673768" @default.
- W3203673768 workType "article" @default.