The square G2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree ∆(G) = 3 we have χ(G2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G2 equals the chromatic number of G2, that is χl(G 2) = χ(G2) for all G. If true, t...

A proper vertex coloring of a graph G is equitable if the size of color classes differ by at most one. The equitable chromatic threshold of G, denoted by ∗Eq(G), is the smallest integer m such that G is equitably n-colorable for all n m. We prove that ∗Eq(G) = (G) if G is a non-bipartite planar graph with girth 26 and (G) 2 or G is a 2-connected outerplanar graph with girth 4. © 2007 Elsevier B...

A linear colouring of a graph is a proper vertex colouring such that the subgraph induced by any two colour classes is a set of vertex disjoint paths. The corresponding linear chromatic number of a graph G, namely lc(G), is the minimum number of colours in a linear colouring of G. We prove that for a graph G with girth g ≥ 8 and maximum degree ∆ ≥ 7 the inequality on its linear colouring number...

The 2-restricted edge-connectivity λ′′ of a graph G is defined to be the minimum cardinality |S| of a set S of edges such thatG−S is disconnected and is of minimum degree at least two. It is known that λ′′ ≤ g(k − 2) for any connected k-regular graph G of girth g other than K4, K5 and K3,3, where k ≥ 3. In this paper, we prove the following result: For a connected vertex-transitive graph of ord...

Let e be a positive integer, p be a odd prime, q = p e, and Fq be the finite field of q elements. Let f2, f3 ∈ Fq[x, y]. The graph G = Gq(f2, f3) is a bipartite graph with vertex partitions P = Fq and L = Fq , and edges defined as follows: a vertex (p) = (p1, p2, p3) ∈ P is adjacent to a vertex [l] = [l1, l2, l3] if and only if p2 + l2 = f2(p1, l1) and p3 + l3 = f3(p1, l1). Motivated by some qu...

We prove tight (up to small constant factors) results on how localized an eigenvector of a high girth regular graph can be (the girth is the length of the shortest cycle). This study was initiated by Brooks and Lindenstrauss [BL13] who relied on the key observation that certain suitably normalized averaging operators on high girth graphs are hyper-contractive (have small `1 → `∞ norm) and can b...

For an integer k ≥ 1, a graph G is k-colorable if there exists a mapping c : VG → {1, . . . , k} such that c(u) 6= c(v) whenever u and v are two adjacent vertices. For a fixed integer k ≥ 1, the k-COLORING problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g ≥ 4 we determine a lower bound `(g), such that ...

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first nontrivial algorithm for the problem, given by Djidjev, runs in O(n5/4 logn) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log n). Weimann and Yuster further reduced the running t...

A graph G is (k, 0)-colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 k+2 is (k, 0)-colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)-colorable. On the othe...