An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ∆ are injectively ∆-colorable. We also show that all planar graphs of girth ≥10 are injectively (...

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distancetransitive graph. As an application of the result, we obtain some examples of weakly distanceregular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distanc...

The aim of this paper is to construct new small regular graphs with girth 7 using integer programming techniques. Over the last two decades solvers for integer programs have become more and more powerful and have proven to be a useful aid for many hard combinatorial problems. Despite successes in many related fields, these optimisation tools have so far been absent in the quest for small regula...

Purpose. To report the mean and median pelvic limb joint angles and girth measurements in miniature Dachshunds presenting with varying degrees of pelvic limb weakness secondary to thoracolumbar intervertebral disc extrusion. Methods. 15 miniature Dachshunds who presented to WSU-VTH for thoracolumbar disc extrusion. Dachshunds varied in neurologic status from ambulatory paraparetic to paraplegic...

Lower bounds on the subdominant eigenvalue of regular graphs of given girth are derived. Our approach is to approximate the discrete spectrum of a finite regular graph by the continuous spectrum of an infinite regular tree. We interpret these spectra as probability distributions and the girth condition as equalities between the moments of these distributions. Then the associated orthogonal poly...

We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (logd−1 |G|) /2 and that random d-regular Cayley graphs of simple algebraic groups over Fq asymptotically almost surely have girth at least logd−1 |G|/ dim(G). For the symmetric p-groups the girth is between log log |G| and (log |G|) with α < 1. Several conjectures and open quest...

A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.

In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most 5, and for the pth chromatic number χp, from which follows in...

In this paper, we give two constructions of weakly distance-regular digraphs of girth 2, and prove that certain quotient digraph of a commutative weakly distancetransitive digraph of girth 2 is a distance-transitive graph. As an application of the result, we not only give some constructions of weakly distance-regular digraphs which are not weakly distance-transitive, but determine a special cla...