W. He et al. showed that a planar graph of girth 11 can be decomposed into a forest and a matching. D. Kleitman et al. proved the same statement for planar graphs of girth 10. We further improve the bound on girth to 9.
Abstract:The M-degree of an edge xy in a graph is the maximum of the degrees of x and y. The M-degree of a graph G is the minimum over M-degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4-cycles has M-degree at most 8 and gave an example of such a graph with M-degree 3. This yields upper bounds on the game chromatic number of C 4 -free planar graphs. We determine the maximum possible M-degrees for
PurposeAn increasing number of human resource development (HRD) theorists and researchers are calling for a broader philosophical framework for HRD within management practice. The concept of workplace spirituality has received significant attention in this context. The purpose of this paper is to discuss the role of religion of Islam in filling this need for a spiritual philosophical framework and to highlight the lessons that can be learned from Islamic traditions. Finally, the authors call for revisiting some of the major motivation theories of HRD.Design/methodology/approachAfter discussing relevant philosophical, spiritual and HRD literature, this paper proposes modification in expectancy theory of motivation.FindingsThis paper emphasizes holistic education and human development in HRD. It proposes an enhanced role for objectives’ valence and value in organizational motivation. It also shows how earlier Islamic traditions had already practised the modern HRD principles.Research limitations/implicationsBeing conceptual and theoretical in nature, the suggested motivation model needs rigor, further testing and empirical analysis.Practical implicationsThe paper suggests that HRD ought to incorporate holistic education and human development as its main drivers. Furthermore, organizations need to put more emphasis on the value of ethical and normative objectives that may involve delayed or reduced gratification.Social implicationsThe paper implies that by giving more emphasis to the value of ethical and moral goals, organizations and human resources would be more responsible to social responsibilities.Originality/valueThe paper proposes a new dimension in the expectancy theory of motivation and also provides justification for the role of spirituality as a philosophical framework in HRD.
For an edge xy, let M (xy) be the maximum of the degrees of x and y. The minimax degree (or M -degree) of a graph G is M * (G) = min{M (xy)|xy ∈ E(G)}. In order to get upper bounds on the game chromatic number of planar graphs, He, Hou, Lih, Shao, Wang, and Zhu showed that every planar graph G without leaves and 4-cycles has minimax degree at most 8, which was improved by Borodin, Kostochka, Sheikh, and Yu to the sharp bound 7. We show that every planar graph G without leaves and 4-and 5-cycles has M -degree at most 5, which bound is sharp. We also show that every planar graph G without leaves and cycles of length from 4 to 7 has M -degree at most 4, which bound is attained even on planar graphs with no cycles of length from 4 to arbitrarily large number. Besides, we give sufficient conditions for a planar graph to have M -degrees 3 and 2. Similar results are obtained for graphs embeddable into the projective plane, the torus and the Klein bottle.
W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.
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