3 List Coloring Graphs of Girth at least Five on Surfaces

Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvorak, Kral and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this result to prove Havel's conjecture that a planar graph whose triangles are pairwise far enough apart is 3-colorable. As for list-coloring, Dvorak proved that a planar graph whose cycles of size at most four are pairwise far enough part is 3-choosable. In this article, we generalize these results. First we prove a linear isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many 4-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a 4-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvorak's result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is 3-choosable. Finally, we generalize to surfaces Thomassen's result that every planar graph of girth at least five has exponentially many distinct 3-list-colorings. Specifically, we show that every graph of girth at least five that has a 3-list-coloring has $2^{\Omega(n)-O(g)}$ distinct 3-list-colorings.Comment: 33 page

Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces

In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first such linear-time algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in S is 5-colorable and similarly in 2003 if a graph of girth at least five embedded in S is 3-colorable. Using the theory of hyperbolic families, the author and Thomas showed such algorithms exist for list-colorings. Dvorak and Kawarabayashi actually gave an $O(n^{O(g+1)})$-time algorithm to find such colorings (if they exist) in n-vertex graphs where g is the Euler genus of the surface. Here we provide the first such algorithm whose exponent does not depend on the genus; indeed, we provide a linear-time algorithm. In 1988, Goldberg, Plotkin and Shannon provided a deterministic distributed algorithm for 7-coloring n-vertex planar graphs in $O(\log n)$ rounds. In 2018, Aboulker, Bonamy, Bousquet, and Esperet provided a deterministic distributed algorithm for 6-coloring n-vertex planar graphs in $O(\log^3 n)$ rounds. Their algorithm in fact works for 6-list-coloring. They also provided an $O(\log^3 n)$-round algorithm for 4-list-coloring triangle-free planar graphs. Chechik and Mukhtar independently obtained such algorithms for ordinary coloring in $O(\log n)$ rounds, which is best possible in terms of running time. Here we provide the first polylogarithmic deterministic distributed algorithms for 5-coloring n-vertex planar graphs and similarly for 3-coloring planar graphs of girth at least five. Indeed, these algorithms run in $O(\log n)$ rounds, work also for list-colorings, and even work on a fixed surface (assuming such a coloring exists).Comment: 20 pages, revised versio

On the Minimum Number of Edges in Triangle-Free 5-Critical Graphs

Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|>= (9/4)|V(G)| - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists e,d > 0 such that if G is a 5-critical graph, then |E(G)| >= (9/4 + e)|V(G)|- 5/4 - dT(G), where T(G) is the maximum number of vertex-disjoint cliques of size three or four where cliques of size four have twice the weight of a clique of size three. As a corollary, a triangle-free 5-critical graph G satisfies: |E(G)|>=(9/4 + e)|V(G)| - 5/4.Comment: 25 pages, revised according to referee comment

Characterizing 4-Critical Graphs of Ore-Degree at most Seven

Dirac introduced the notion of a k-critical graph, a graph that is not (k-1)-colorable but whose every proper subgraph is (k-1)-colorable. Brook's Theorem states that every graph with maximum degree k is k-colorable unless it contains a subgraph isomorphic to K_{k+1} (or an odd cycle for k=2). Equivalently, for all k>=4, the only k-critical graph of maximum degree k-1 is K_k. A natural generalization of Brook's theorem is to consider the Ore-degree of a graph, which is the maximum of d(u)+d(v) over all edges uv. Kierstead and Kostochka proved that for all k>=6 the only k-critical graph with Ore-degree at most 2k-1 is K_k. Kostochka, Rabern and Steibitz proved that the only 5-critical graphs with Ore-degree at most 9 are K_5 and a graph they called O_5. A different generalization of Brook's theorem, motivated by Hajos' construction, is Gallai's conjectured bound on the minimum density of a k-critical graph. Recently, Kostochka and Yancey proved Gallai's conjecture. Their proof for k>=5 implies the above results on Ore-degree. However, the case for k=4 remains open, which is the subject of this paper. Kostochka and Yancey's short but beautiful proof for the case k=4 says that if $G$ is a $4$-critical graph, then |E(G)|>= (5|V(G)|-2)/3. We prove the following bound which is better when there exists a large independent set of degree three vertices: if G is a 4-critical graph G, then |E(G)|>= 1.6 |V(G)| + .2 alpha(D_3(G)) - .6, where D_3(G) is the graph induced by the degree three vertices of G. As a corollary, we characterize the 4-critical graphs with Ore-degree at most seven as precisely the graphs of Ore-degree seven in the family of graphs obtained from K_4 and Ore compositions.Comment: 35 page

The List Linear Arboricity of Graphs

A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating that every $d$-regular graph $G$ has linear arboricity $\lceil \frac{d+1}{2} \rceil$. In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC was raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Arboricity Conjecture holds asymptotically.Comment: 17 page

Fractional coloring with local demands

We investigate fractional colorings of graphs in which the amount of color given to a vertex depends on local parameters, such as its degree or the clique number of its neighborhood; in a \textit{fractional $f$-coloring}, vertices are given color from the $[0, 1]$-interval and each vertex $v$ receives at least $f(v)$ color. By Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers. However, these problems are most natural in the framework of fractional coloring, and the concept of coloring is crucial to most of our proofs. Our results and conjectures considerably generalize many well-known fractional coloring results, such as the fractional relaxation of Reed's Conjecture, Brooks' Theorem, and Vizing's Theorem. Our results also imply previously unknown bounds on the independence number of graphs. We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has a fractional $f$-coloring or $G$ contains a clique $K$ such that $\sum_{v\in K}f(v) > 1$. This result generalizes the famous Caro-Wei Theorem, and it implies that every graph $G$ with no simplicial vertex has an independent set of size at least $\sum_{v\in V(G)}\frac{1}{d(v) + 1/2}$, which is tight for the 5-cycle.Comment: 37 pages including 4 page Appendi

Exponentially Many 4-List-Colorings of Triangle-Free Graphs on Surfaces

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. Postle and Thomas proved that if $G$ is a graph on $n$ vertices embedded on a surface $\Sigma$ of genus $g$, then there exist constants $\epsilon,c_g > 0$ such that if $G$ has an $L$-coloring, then $G$ has at least $c_g2^{\epsilon n}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ or if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. More generally, they proved that there exist constants $\epsilon,\alpha>0$ such that if $G$ is a graph on $n$ vertices embedded in a surface $\Sigma$ of fixed genus $g$, $H$ is a proper subgraph of $G$, and $\phi$ is an $L$-coloring of $H$ that extends to an $L$-coloring of $G$, then $\phi$ extends to at least $2^{\epsilon(n - \alpha(g + |V(H)|))}$ distinct $L$-colorings of $G$ if $L$ is a 5-list-assignment or if $L$ is a 3-list-assignment and $G$ has girth at least five. We prove the same result if $G$ is triangle-free and $L$ is a 4-list-assignment of $G$, where $\epsilon=\frac{1}{8}$, and $\alpha= 130$.Comment: 12 pages, 2 figure

Density of 5/2-critical graphs

A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C_5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n>=4 vertices has at least (5n-2)/4 edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.Comment: 26 pages, 3 figure

Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8

We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring. Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.Comment: 22 pages, 3 figures; v2: improves presentatio

On the Minimum Edge-Density of 4-Critical Graphs of Girth Five

We prove that if G is a 4-critical graph of girth at least five then |E(G)|>=(5|V(G)|+2)/3. As a corollary, graphs of girth at least five embeddable in the Klein bottle or torus are 3-colorable. These are results of Thomas and Walls, and Thomassen respectively. The proof uses the new potential technique developed by Kostochka and Yancey who proved that 4-critical graphs satisfy: |E(G)|>=(5|V(G)|-2)/3.Comment: 17 page