Seog-Jin Kim

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

Counterexamples to the List Square Coloring Conjecture

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

Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree

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

List dynamic coloring of sparse graphs

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

Graphs having many holes but with small competition numbers

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

Chromatic-choosability of the power of graphs

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

A sufficient condition for DP-4-colorability

, and χi(G)=6.

Improved bounds on the chromatic numbers of the square of Kneser graphs

The square G2 of a graph G is the graph defined on VG such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Let i¾?H and i¾?i¾?H be the chromatic number and the list chromatic number of a graph H, respectively. A graph H is called chromatic-choosable if i¾?i¾?H=i¾?H. It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall Choosability conjectures and multicircuits, Discrete Math., 240 2001, 123-143 conjectured that i¾?i¾?G2=i¾?G2 for every graph G, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value i¾?i¾?G2-i¾?G2 can be arbitrarily large.

Decomposition of Sparse Graphs into Forests: The Nine Dragon Tree Conjecture for k ≤ 2

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of over all subgraphs H with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if , then G decomposes into forests with one having maximum degree at most d. The conjecture was previously proved for ; we prove it for and when and . For , we can further restrict one forest to have at most two edges in each component. For general , we prove weaker conclusions. If , then implies that G decomposes into k forests plus a multigraph (not necessarily a forest) with maximum degree at most d. If , then implies that G decomposes into forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs.

Coloring the square of graphs whose maximum average degree is less than 4

A dynamic coloring of a graph G is a proper coloring of the vertex set V (G) such that each vertex neighborhood of size at least 2 receives at least two distinct colors. The list dynamic chromatic number chd(G) of G is the least integer k such that for every list assignment of size k to each vertex of G, there is a dynamic coloring of G such that each vertex is colored by a color from its list. We proved that chd(G) ≤ 4 if Mad(G) < 8/3 where Mad(G) is the maximum average degree of G. And chd(G) ≤ 4 if G is a planar graph of girth at least 7. Both results are sharp. In addition, we show that chd(G) ≤ 6 for every planar graph G.