The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove tha...The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.展开更多
In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this proble...In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this problem for n sufficiently large.展开更多
For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been wid...For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been widely studied. Moreover, the form of minimum such an f is also concerned. A result of Schiermeyer shows that every -free graph G with clique number has . Chudnovsky and Sivaraman proved that every -free with clique number graph is -colorable. In this paper, for any -free graph G with clique number , we prove that . The main methods in the proof are set partition and induction.展开更多
The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum deg...The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.展开更多
Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has...Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ? 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.展开更多
By leveraging the 5G enabled vehicular ad hoc network(5G-VANET), it is widely recognized that connected vehicles have the potentials to improve road safety, transportation intelligence and provide in-vehicle entertain...By leveraging the 5G enabled vehicular ad hoc network(5G-VANET), it is widely recognized that connected vehicles have the potentials to improve road safety, transportation intelligence and provide in-vehicle entertainment experience. However, many enabling applications in 5G-VANET rely on the efficient content sharing among mobile vehicles, which is a very challenging issue due to the extremely large data volume, rapid topology change, and unbalanced traffic. In this paper, we investigate content prefetching and distribution in 5G-VANET. We first introduce an edge computing based hierarchical architecture for efficient distribution of large-volume vehicular data. We then propose a multi-place multi-factor prefetching scheme to meet the rapid topology change and unbalanced traffic. The content requests of vehicles can be served by neighbors, which can improve the sharing efficiency and alleviate the burden of networks. Furthermore, we use a graph theory based approach to solve the content distribution by transforming it into a maximum weighted independent set problem. Finally, the proposed scheme is evaluated with a greedy transmission strategy to demonstrate its efficiency.展开更多
A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with...A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ（G） ≥ 7 and without chordal 7-cycles,then G has a（Δ（G） ＋ 1）-total-coloring.展开更多
文摘The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.
基金support from FCT—Fundacao para a Ciencia e a Tecnologia(Portugal),through Projects PTDC/MAT/113207/2009PEst-OE/MAT/UI0297/2011(CMA).
文摘In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this problem for n sufficiently large.
文摘For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been widely studied. Moreover, the form of minimum such an f is also concerned. A result of Schiermeyer shows that every -free graph G with clique number has . Chudnovsky and Sivaraman proved that every -free with clique number graph is -colorable. In this paper, for any -free graph G with clique number , we prove that . The main methods in the proof are set partition and induction.
基金supported by Natural Science Foundation of Shandong Province (Grant No. ZR2009AM009)Scientific Research Foundation for the Excellent Middle-Aged and Youth Scientists of Shandong Province (Grant No. BS2012SF016)National Natural Science Foundation of China (Grant Nos.11001055 and 11101243)
文摘The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 10471131)
文摘Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ? X ve (G) ? δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ? 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.
基金the support of National Science and Technology Major Project of the Ministry of Science and Technology of China under Grant No.2016ZX03001025003the Natural Science Foundation of Beijing under Grant No.4181002+2 种基金the Natural Science Foundation of China under Grant No.91638204BUPT Excellent Ph.D. Students Foundation under Grant No.CX2018210Natural Sciences and Engineering Research Council (NSERC),Canada
文摘By leveraging the 5G enabled vehicular ad hoc network(5G-VANET), it is widely recognized that connected vehicles have the potentials to improve road safety, transportation intelligence and provide in-vehicle entertainment experience. However, many enabling applications in 5G-VANET rely on the efficient content sharing among mobile vehicles, which is a very challenging issue due to the extremely large data volume, rapid topology change, and unbalanced traffic. In this paper, we investigate content prefetching and distribution in 5G-VANET. We first introduce an edge computing based hierarchical architecture for efficient distribution of large-volume vehicular data. We then propose a multi-place multi-factor prefetching scheme to meet the rapid topology change and unbalanced traffic. The content requests of vehicles can be served by neighbors, which can improve the sharing efficiency and alleviate the burden of networks. Furthermore, we use a graph theory based approach to solve the content distribution by transforming it into a maximum weighted independent set problem. Finally, the proposed scheme is evaluated with a greedy transmission strategy to demonstrate its efficiency.
基金Supported by National Natural Science Foundation of China(Grant No.11271006)
文摘A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ（G） ≥ 7 and without chordal 7-cycles,then G has a（Δ（G） ＋ 1）-total-coloring.
