Wang and Lih conjectured that for every g ≥ 5, there exists a number M(g) such that the chromatic number of the square of every planar graph of girth at least g and maximum degree ∆ ≥ M(g) is ∆ + 1. We disprove the conjecture for g ∈ {5, 6} and prove the existence of the number M(g) for g ≥ 7. More generally, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2dp/q...

In this article, some structures in the projective plane of order q are found which allow us to construct small k regular balanced bipartite graphs of girth 6 for all k ≤ q . When k = q , the order of these q-regular graphs is 2(q2−1); andwhen k ≤ q−1, the order of these k -regular graphs is 2(qk − 2). Moreover, the incidence matrix of a k -regular balanced bipartite graph of girth 6 having 2(q...

in a prospective descriptive study, the usefulness of symphysis-fundal height and the product of abdominal girth and fundal height in predicting birth weight < 2500 g and > 4000 g were examined. Fundal height and abdominal girth were measured at the time of admission on a sample of 795 parturient women at ateaching hospital in the Islamic Republic of Iran. Receiver operating characteristics cur...

We present a local algorithm producing an independent set of expected size 0.44533n on large-girth 3-regular graphs and 0.40407n on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with 1.34105n edges on large-girth 3regular graphs. These decrease the gaps between the best known upper and lower bounds from 0.0178 to 0.01, from 0.0242 to 0.0123 and from ...

Let C and D be digraphs. A mapping f : V (D) → V (C) is a Ccolouring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u) = f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely Ccolourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphis...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

Let 2 ≤ r < m and g ≥ 4 even be three positive integers. A graph with a degree set {r,m}, girth g and minimum order is called a bi-regular cage or an ({r,m}; g)-cage, and its order is denoted by n({r,m}; g). In this paper we obtain constructive upper bounds on n({r,m}; g) for some values of r,m and even girth at least 8.

Reed conjectured that for every > 0 and ∆ there exists g such that the fractional total chromatic number of a graph with maximum degree ∆ and girth at least g is at most ∆ + 1 + . We prove the conjecture for ∆ = 3 and for even ∆ ≥ 4 in the following stronger form: For each of these values of ∆, there exists g such that the fractional total chromatic number of any graph with maximum degree ∆ and...

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k 12(g+1) vertices whose maximal degree is at most 5k 13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G) log jGj 13 log k and maximum degree (G) 6k 13 can exist. We also study several related problems.

In this paper we introduce efficient parallel algorithms for finding the girth in a graph or digraph, where girth is the length of a shortest cycle. We empirically compare our algorithms by using two common APIs for parallel programming in C++, which are OpenMP for multiple CPUs and CUDA for multi-core GPUs. We conclude that both hardware platforms and programming models have their benefits.