A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic numbe...A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic number will be determined for the Mycielski graphs of trees using the method of induction.展开更多
Let G be a simple graph with no isolated edge. An Ⅰ-total coloring of a graph G is a mapping φ : V(G) ∪ E(G) → {1, 2, · · ·, k} such that no adjacent vertices receive the same color and no adjacent ...Let G be a simple graph with no isolated edge. An Ⅰ-total coloring of a graph G is a mapping φ : V(G) ∪ E(G) → {1, 2, · · ·, k} such that no adjacent vertices receive the same color and no adjacent edges receive the same color. An Ⅰ-total coloring of a graph G is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, we have C_φ(u) = C_φ(v), where C_φ(u) denotes the set of colors of u and its incident edges. The minimum number of colors required for an adjacent vertex distinguishing Ⅰ-total coloring of G is called the adjacent vertex distinguishing Ⅰ-total chromatic number, denoted by χ_at^i(G).In this paper, we characterize the adjacent vertex distinguishing Ⅰ-total chromatic number of outerplanar graphs.展开更多
An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence ...An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.展开更多
IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas.In this paper,we discu...IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas.In this paper,we discuss interference models and address how they can affect the design of channel assignment in rural mesh networks.We present a new channel assignment framework based on graph coloring for rural wireless mesh networks.The goal of the framework is to allow synchronously transmitting or receiving data from multiple neighbor links at the same time,and continuously doing full-duplex data transfer on every link,creating an efficient rural mesh network without interference.Channel assignment is shown to be NP-hard.We frame this channel allocation problem in terms of Adjacent Vertex Distinguishing Edge Coloring(AVDEC).Detailed assignment results on grid topology are presented and discussed.Furthermore,we design an algorithm.Finally,we evaluate the performance of the proposed algorithm through extensive simulations and show the algorithm is effective to the regular grid topologies,and the number of colors used by the algorithm is upper bounded by+1.Hence the algorithm guarantees that the number of channels available in standards such as IEEE802.11a is sufficient to have a valid AVDEC for many grid topologies.We also evaluate the proposed algorithm for arbitrary graphs.The algorithm provides a lower upper bound on the minimum number of channels to the AVDEC index channel assignment problem.展开更多
A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges ...A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to v, where uv ∈E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ'αα(G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ'αα(G)≤32△.展开更多
Let G = (V, E) be a simple connected graph, and |V(G)| ≥ 2. Let f be a mapping from V(G) ∪ E(G) to {1,2…, k}. If arbitary uv ∈ E(G),f(u) ≠ f(v),f(u) ≠ f(uv),f(v) ≠ f(uv); arbitary uv, uw...Let G = (V, E) be a simple connected graph, and |V(G)| ≥ 2. Let f be a mapping from V(G) ∪ E(G) to {1,2…, k}. If arbitary uv ∈ E(G),f(u) ≠ f(v),f(u) ≠ f(uv),f(v) ≠ f(uv); arbitary uv, uw ∈ E(G)(v ≠ w), f(uv) ≠ f(uw);arbitary uv ∈ E(G) and u ≠ v, C(u) ≠ C(v), whereC(u)={f(u)}∪{f(uv)|uv∈E(G)}.Then f is called a k-adjacent-vertex-distinguishing-proper-total coloring of the graph G(k-AVDTC of G for short). The number min{k|k-AVDTC of G} is called the adjacent vertex-distinguishing total chromatic number and denoted by χat(G). In this paper we prove that if △(G) is at least a particular constant and δ ≥32√△ln△, then χat(G) ≤ △(G) + 10^26 + 2√△ln△.展开更多
基金Foundation item: Supported by Natural Science Foundation of China(60503002)
文摘A k-proper total coloring of G is called adjacent distinguishing if for any two adjacent vertices have different color sets. According to the property of trees, the adjacent vertex distinguishing total chromatic number will be determined for the Mycielski graphs of trees using the method of induction.
基金Supported by the National Natural Science Foundation of China(61163037,61163054,61363060)
文摘Let G be a simple graph with no isolated edge. An Ⅰ-total coloring of a graph G is a mapping φ : V(G) ∪ E(G) → {1, 2, · · ·, k} such that no adjacent vertices receive the same color and no adjacent edges receive the same color. An Ⅰ-total coloring of a graph G is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, we have C_φ(u) = C_φ(v), where C_φ(u) denotes the set of colors of u and its incident edges. The minimum number of colors required for an adjacent vertex distinguishing Ⅰ-total coloring of G is called the adjacent vertex distinguishing Ⅰ-total chromatic number, denoted by χ_at^i(G).In this paper, we characterize the adjacent vertex distinguishing Ⅰ-total chromatic number of outerplanar graphs.
基金Supported by the State Ethnic Affairs Commission of China (Grant No. 08XB07)
文摘An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.
基金Supported by the National Natural Science Foundation of China(No.71231004 and No.61004086)
文摘IEEE 802.11 based wireless mesh networks with directional antennas are expected to be a new promising technology and an economic approach for providing wireless broadband services in rural areas.In this paper,we discuss interference models and address how they can affect the design of channel assignment in rural mesh networks.We present a new channel assignment framework based on graph coloring for rural wireless mesh networks.The goal of the framework is to allow synchronously transmitting or receiving data from multiple neighbor links at the same time,and continuously doing full-duplex data transfer on every link,creating an efficient rural mesh network without interference.Channel assignment is shown to be NP-hard.We frame this channel allocation problem in terms of Adjacent Vertex Distinguishing Edge Coloring(AVDEC).Detailed assignment results on grid topology are presented and discussed.Furthermore,we design an algorithm.Finally,we evaluate the performance of the proposed algorithm through extensive simulations and show the algorithm is effective to the regular grid topologies,and the number of colors used by the algorithm is upper bounded by+1.Hence the algorithm guarantees that the number of channels available in standards such as IEEE802.11a is sufficient to have a valid AVDEC for many grid topologies.We also evaluate the proposed algorithm for arbitrary graphs.The algorithm provides a lower upper bound on the minimum number of channels to the AVDEC index channel assignment problem.
基金Supported by the Natural Science Foundation of Gansu Province(3ZS051-A25-025)
文摘A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to v, where uv ∈E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ'αα(G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ'αα(G)≤32△.
基金the Natural Science Foundation of Gansu Province (No. 3ZS051-A25-025) the Foundation of Gansu Provincial Department of Education (No. 0501-03).
文摘Let G = (V, E) be a simple connected graph, and |V(G)| ≥ 2. Let f be a mapping from V(G) ∪ E(G) to {1,2…, k}. If arbitary uv ∈ E(G),f(u) ≠ f(v),f(u) ≠ f(uv),f(v) ≠ f(uv); arbitary uv, uw ∈ E(G)(v ≠ w), f(uv) ≠ f(uw);arbitary uv ∈ E(G) and u ≠ v, C(u) ≠ C(v), whereC(u)={f(u)}∪{f(uv)|uv∈E(G)}.Then f is called a k-adjacent-vertex-distinguishing-proper-total coloring of the graph G(k-AVDTC of G for short). The number min{k|k-AVDTC of G} is called the adjacent vertex-distinguishing total chromatic number and denoted by χat(G). In this paper we prove that if △(G) is at least a particular constant and δ ≥32√△ln△, then χat(G) ≤ △(G) + 10^26 + 2√△ln△.