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- Landon Rabern
- SIAM J. Discrete Math.
- 2008

In [5], Reed conjectures that every graph satisfies Ï‡ â‰¤ âŒˆÏ‰+âˆ†+1 2 âŒ‰ . We prove this holds for graphs with disconnected complement. Combining this fact with a result of Molloy proves the conjecture forâ€¦ (More)

- Landon Rabern
- 2011

We prove thatKÏ‡(G) is the only vertex critical graphG with Ï‡(G) â‰¥ âˆ†(G) â‰¥ 6 and Ï‰(H(G)) â‰¤ âŒŠ âˆ†(G) 2 âŒ‹ âˆ’ 2. Here H(G) is the subgraph of G induced on the vertices of degree at least Ï‡(G). Settingâ€¦ (More)

- Landon Rabern
- Journal of Graph Theory
- 2011

Satisfaction. Given a relation R between graph functions we say that a graph G satisfies R if plugging G into each function gives a true statement (e.g. G satisfies Ï‡ â‰¥ âˆ† means Ï‡(G) â‰¥ âˆ†(G)). We proveâ€¦ (More)

- Landon Rabern
- Journal of Graph Theory
- 2014

It is shown that any graph with maximum degree âˆ† in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds 6k 2 6k2+1 âˆ† + k+ 6 is either (âˆ† âˆ’ k)-colorableâ€¦ (More)

- Alexandr V. Kostochka, Landon Rabern, Michael Stiebitz
- Discrete Mathematics
- 2012

Let G be a color-critical graph with Ï‡(G) â‰¥ Î”(G) = 2t + 1 â‰¥ 5 such that the subgraph of G induced by the vertices of degree 2t+1 has clique number at most tâˆ’1. We prove that then either t â‰¥ 3 and G =â€¦ (More)

We prove that every k-list-critical graph (k â‰¥ 7) on n â‰¥ k + 2 vertices has at least 1 2 ( k âˆ’ 1 + kâˆ’3 (kâˆ’c)(kâˆ’1)+kâˆ’3 ) n edges where c = (k âˆ’ 3) ( 1 2 âˆ’ 1 (kâˆ’1)(kâˆ’2) ) . This improves the boundâ€¦ (More)

- Landon Rabern
- Electr. J. Comb.
- 2011

We prove that if G is the line graph of a multigraph, then the chromatic number Ï‡(G) of G is at most max { Ï‰(G), 7âˆ†(G)+10 8 } where Ï‰(G) and âˆ†(G) are the clique number and the maximum degree of G,â€¦ (More)

- Landon Rabern
- Electr. J. Comb.
- 2007

Way back in 1977, Borodin and Kostochka made the following conjecture (see [1]). Conjecture. Every graph satisfying Ï‡ â‰¥ âˆ† â‰¥ 9 contains a Kâˆ†. Examples exist showing that the âˆ† â‰¥ 9 condition isâ€¦ (More)

- Landon Rabern
- SIAM J. Discrete Math.
- 2006

We introduce a notion of vertex association and consider sequences of these associations. This allows for slick proofs of a few known theorems as well as showing that for any induced subgraph H of G,â€¦ (More)

- Landon Rabern
- Discrete Mathematics
- 2014

We characterize the initial positions from which the first player has a winning strategy in a certain two-player game. This provides a generalization of Hallâ€™s Theorem. Vizingâ€™s Theorem onâ€¦ (More)