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Given a graph G and a positive integer p, χ p (G) is the minimum number of colours needed to colour the vertices of G so that for any i ≤ p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χ p (G) in terms of the k-colouring number col k (G) of G for k = 2 p−2. Conversely, for each integer k, we also prove an upper bound for col k (G) in terms of χ k+2 (G). As a consequence, for a class K of graphs, the following two statements are equivalent: (a) For every positive integer p, χ p (G) is bounded by a constant for all G ∈ K.

This paper discusses a variation of the game chromatic number of a graph: the game coloring number. This parameter provides an upper bound for the game chromatic number of a graph. We show that the game coloring number of a planar graph is at most 19. This implies that the game chromatic number of a planar graph is at most 19, which improves the previous known upper bound for the game chromatic number of planar graphs. Academic Press

BackgroundThe purpose of this study was to compare the clinical outcomes of elderly patients undergoing surgery for treatment of unstable trochanteric fractures receiving either proximal femoral nails anti-rotation-Asia (PFNA-IIs) or InterTan nails (ITs).MethodsBetween January 1, 2012, and June 31, 2015, 168 elderly patients with unstable intertrochanteric femur fractures enrolled in this study. The only intervention was ITs or PFNA-IIs of the unstable trochanteric femur fractures. Follow-up was at 1, 3, 6, and 12 months postoperatively and yearly thereafter. Intraoperative variables and postoperative complications were compared between the two groups.ResultsEight patients died, six were too infirmed for follow-up, and seven were lost during follow-up, leaving 147 patients meeting the criteria were evaluated at a mean follow-up of 20 months (range 16–26 months). Significant differences were observed between the two groups regarding local complications (IT, n = 10 vs. PFNA-II, n = 20), varus collapse of the head/neck or femoral shaft fractures at the tip of the nail (IT, n = 1 vs. PFNA-II, n = 8), femoral neck shortening (IT, 4.4 ± 1.1 mm vs. PFNA-II, 7.4 ± 2.4 mm), fracture healing time (IT, 14.7 ± 2.1 weeks vs. PFNA-II, 15.7 ± 2.4 weeks), femoral shaft fractures (IT, n = 0 vs. PFNA-II, n = 4), rotational loss of reduction (IT, n = 0 vs. PFNA-II, n = 9), lateral cortex fractures of the proximal femur or lateral greater trochanter fractures (IT, n = 8 vs. PFNA-II, n = 1), operative time (IT, 71.9 ± 6.8 min vs. PFNA-II, 52.3 ± 4.0 min), intraoperative blood loss (IT, 190.6 ± 6.0 mL vs. PFNA-II, 180.9 ± 10.8 mL), fluoroscopy time (IT, 5.0 ± 0.48 min vs. PFNA-II, 2.8 ± 0.33 min), hospital stay (IT, 9.65 ± 0.95 days vs. PFNA-II, 8.58 ± 0.93 days), cut-out (IT, n = 0 vs. PFNA-II, n = 6), and tip-apex distance (IT, 26.7 ± 0.91 mm vs. PFNA-II, 23.2 ± 1.22 mm). No significant differences existed for the other observation indexes (p > 0.05).ConclusionsThe IT nail may have more advantage for patients with unstable intertrochanteric fractures of the femur. However, for those complicated with lateral greater trochanter fractures, lateral cortex fractures of the proximal femurs, or unfit for surgery, the PFNA-II nail could be a good option. In addition, a large-sample, multicenter observational study is required for evaluation of its long-term efficacy, and optimal management strategies for specific unstable fracture patterns, different sorts of bone quality, and different levels of patient demand.