Daniel W. Cranston

Note: Strong edge-coloring of graphs with maximum degree 4 using 22 colors

In 1985, Erdos and Nesetril conjectured that the strong edge-coloring number of a graph is bounded above by 54@D^2 when @D is even and 14(5@D^2-2@D+1) when @D is odd. They gave a simple construction which requires this many colors. The conjecture has been verified for @D=<3. For @D=4, the conjectured bound is 20. Previously, the best known upper bound was 23 due to Horak. In this paper we give an algorithm that uses at most 22 colors.

Injective Colorings of Graphs with Low Average Degree

Let mad (G) denote the maximum average degree (over all subgraphs) of G and let χi(G) denote the injective chromatic number of G. We prove that if Δ≥4 and

Note: Short proofs for cut-and-paste sorting of permutations

\mathrm{mad}(G)<\frac{14}{5}

Regular Graphs of Odd Degree Are Antimagic

, then χi(G)≤Δ+2. When Δ=3, we show that

The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs

\mathrm{mad}(G)<\frac{36}{13}

List-Coloring Claw-Free Graphs with

implies χi(G)≤5. In contrast, we give a graph G with Δ=3,

\Delta-1

\mathrm{mad}(G)=\frac{36}{13}

Colors

, and χi(G)=6.

Sufficient sparseness conditions for G 2 to be ( Δ + 1 ) -choosable, when Δ ≥ 5

We consider the problem of determining the maximum number of moves required to sort a permutation of [n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n] can be transformed to the identity in at most @?2n/3@? such moves and that some permutations require at least @?n/2@? moves.

The fractional chromatic number of the plane

An antimagic labeling of a graph G with m edges is a bijection from EG to {1,2,...,m} such that for all vertices u and v, the sum of labels on edges incident to u differs from that for edges incident to v. Hartsfield and Ringel conjectured that every connected graph other than the single edge K2 has an antimagic labeling. We prove this conjecture for regular graphs of odd degree.