Abstract. The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F ) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j).
Total knee arthroplasty (TKA) is a commonly performed procedure that provides reliable pain relief to those suffering from knee arthritis. While technologies and surgical techniques have improved, complications can and do still occur. [1][2][3][4][5][6][7][8][9] One of the most serious potential complications is postoperative patellar instability. 4,10-12 Patellar instability can often be treated by correcting the rotation of malpositioned implants. When the implants are in appropriate position, this can be a more complex problem to treat. Recently, there has been interest in reconstructing the medial patellofemoral ligament (MPFL) to treat postoperative patellar instability. 13,14 The purpose of this case report is to describe the use of MPFL reconstruction as an adjunct to femoral component revision and to treat such a case of postoperative patellar instability. Case ReportA 65-year-old male presented to the clinic with left knee pain. He denied any episodes of trauma and described his pain as being located mostly at his medial compartment. His clinical examination revealed a small effusion and a range of motion from 10 to 115 degrees. He had a negative Lachman test and posterior drawer test and was stable to varus and valgus stress at 0 and 30 degrees. His patella tracked normally with some crepitus. There was no J sign and no lateral tilt of his patella on examination. His patella was neither subluxable nor dislocatable on examination and had a firm end point to lateral push. His radiographs were consistent with tricompartmental osteoarthritis (►Fig. 1). After failing conservative treatment options, he elected to proceed with a TKA.The patient was taken to the operating room and a cruciate retaining TKA (Biomet, Warsaw, IN) was performed without complication. A subvastus approach was used and the patella resurfaced. The patient had a routine postoperative course until 6 weeks postoperatively when he began to complain of patella maltracking and pain after a fall. Clinical examination and radiographs revealed a laterally dislocated patella. The tibial and femoral components were carefully examined for evidence of malrotation, but no obvious malrotation was identified. A computed tomogaphy scan was also used to evaluate the rotational alignment of the components and revealed no evidence of malrotation. Treatment options were discussed and a release of the lateral retinaculum and medial reefing was performed. These were performed in a limited open fashion. The patient was placed in a knee immobilizer for 2 weeks following surgery and then began standard postarthroplasty physical therapy consisting of range of motion exercises, strengthening, and gait training. Upon discontinuation of the knee immobilizer, he was again found to have recurrent lateral patellar dislocations. Treatment options were again discussed and the patient decided to proceed with exploration of his knee arthroplasty with likely revision if malrotated components were found. In Keywords ► patella instability ► arthroplasty ► medial patellofemor...
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