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- W2952395211 abstract "In 1989, Lescure and Meyniel proved, for $d=5, 6$, that every $d$-chromatic graph contains an immersion of $K_d$, and in 2003 Abu-Khzam and Langston conjectured that this holds for all $d$. In 2010, DeVos, Kawarabayashi, Mohar, and Okamura proved this conjecture for $d = 7$. In each proof, the $d$-chromatic assumption was not fully utilized, as the proofs only use the fact that a $d$-critical graph has minimum degree at least $d - 1$. DeVos, Dvov{r}'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture that a graph with minimum degree $d-1$ has an immersion of $K_d$ fails for $d=10$ and $dgeq 12$ with a finite number of examples for each value of $d$, and small chromatic number relative to $d$, but it is shown that a minimum degree of $200d$ does guarantee an immersion of $K_d$. In this paper we show that the stronger conjecture is false for $d=8,9,11$ and give infinite families of examples with minimum degree $d-1$ and chromatic number $d-3$ or $d-2$ that do not contain an immersion of $K_d$. Our examples can be up to $(d-2)$-edge-connected. We show, using Haj'os' Construction, that there is an infinite class of non-$(d-1)$-colorable graphs that contain an immersion of $K_d$. We conclude with some open questions, and the conjecture that a graph $G$ with minimum degree $d - 1$ and more than $frac{|V(G)|}{1+m(d+1)}$ vertices of degree at least $md$ has an immersion of $K_d$." @default.
- W2952395211 created "2019-06-27" @default.
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- W2952395211 date "2012-06-07" @default.
- W2952395211 modified "2023-09-27" @default.
- W2952395211 title "Constructing graphs with no immersion of large complete graphs" @default.
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