The general vertex-distinguishing total chromatic number of a graph G is the minimum integer k, for which the vertices and edges of G are colored using k colors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value of this chromatic number of some special graphs and propose a conjecture on the upper bound of this chromatic number.
1. Introduction
All graphs considered in this paper are simple and finite. For a graph G, we denote by V(G), E(G), Δ(G), and δ(G) the sets of vertices, edges, maximum degree, and minimum degree of G, respectively. For a vertex v of G, dG(v) is the degree of v in G. For any V′⊆V(G), we use G[V′] to denote the subgraph induced by V′. For any undefined terms, the reader is referred to the book [1].
The coloring problem of graphs is one of the classical research areas in graph theory. It has been widely applied to various fields, such as large scheduling [2], assignment of radio frequency [3], and separating combustible chemical combinations [4]. Due to its extensive application, many new variants of colorings have been studied [5].
Recall that a k-edge coloring of a graph G is a mapping f: E(G)→C, where C is a set of k colors. An edge coloring is proper if adjacent edges receive distinct colors. In 1985, Harary and Plantholt [6] first considered point-distinguishing chromatic index, which is a variant of edge coloring. After that, many other variants of edge coloring were introduced, such as vertex-distinguishing proper edge coloring [7], adjacent vertex-distinguishing edge coloring [8], and general adjacent vertex-distinguishing edge coloring [9].
A total k-coloring of a graph G is a coloring of V(G)∪E(G) using k colors. A total k-coloring is proper if no two adjacent or incident elements receive the same color. The minimum number of colors required for a proper total coloring of G is called the total chromatic number of G and is denoted by χt(G). Behzad [10] and Vizing [11] independently made the conjecture that, for any graph G,
(1)χt(G)≤(G)+2.
This is known as the total coloring conjecture (TCC) and is still unproven.
Let f be a total k-coloring of G. The total color set (with respect to f) of a vertex v∈V(G) is the set, denoted by Cf(v), of colors of v and its incident edges. We denote by Cf(G) the set of total color sets of all vertices of G. Furthermore, let S be a subset of V(G)∪E(G); we use Cf(S) to denote the set of colors of elements of S.
Like edge coloring, total coloring also has some variants. In 2005, Zhang et al. [12] added a restriction to the definition of total coloring and proposed a new type of coloring defined as follows.
Definition 1.
Let f be a proper total k-coloring of a graph G. If, for all u,v∈V(G), Cf(u)≠Cf(v), then f is called an adjacent vertex-distinguishing total k-coloring of G, or a k-AVDTC of G for short. The minimum number k for which G has a k-AVDTC is the adjacent vertex-distinguishing total chromatic number of G, denoted by χat(G).
Zhang et al. [12] conjectured that, for any graph G, it has
(2)χat(G)≤Δ(G)+3.
In [13–15], authors proved that there exists a 6-AVDTC of graphs with Δ=3, which indicates conjecture (2) holds for such graphs. For further research on adjacent vertex-distinguishing total chromatic number, one may refer to [16–23].
For a k-AVDTCf of a graph G, if Cf(u)≠Cf(v) is required for any two distinct vertices u,v, then f is called a vertex-distinguishing total k-coloring of G, abbreviated as k-VDTC. The minimum number k such that G has a k-VDTC is called the vertex-distinguishing total chromatic number, denoted by χvt(G) [24]. Zhang et al. conjectured in [24] that, for any graph G, it follows that
(3)μt(G)≤χvt(G)≤μt(G)+1,
where μt(G)=min{k∣(ki+1)≥ni,δ≤i≤Δ}.
In this paper, we introduce a variant of vertex-distinguishing total coloring of a graph G, which relaxes the restriction that the coloring is proper. We now present the detailed definition as follows.
Definition 2.
Let G be a graph and k be a positive integer. A total coloring f of G using k colors is called a general vertex-distinguishing total k-coloring of G(or k-GVDTC of G briefly) if, for all u,v∈V(G), Cf(u)≠Cf(v). The minimum number k for which G has a k-GVDTC is the general vertex-distinguishing total chromatic number, denoted by χgvt(G).
It is evident that χgvt(G) does exist for any graph G. In this paper, we study the general vertex-distinguishing total coloring of some special classes of graphs and obtain the exact value of the general vertex-distinguishing total chromatic number of these graphs. Furthermore, we propose a conjecture on the upper bound of general vertex-distinguishing total chromatic number of a graph.
2. Main Results
We first present a trivial lower bound on the general vertex-distinguishing total chromatic number of a graph.
Theorem 3.
Let G be a graph on n vertices. Then
(4)χgvt(G)≥⌈log2(n+1)⌉.
Proof.
Let χgvt(G)=k. It follows that n≤(k1)+(k2)+⋯+(kk)=2k-1, so k≥⌈log2(n+1)⌉.
Notice that the lower bound of Theorem 3 can be attained in graphs, such as the n-vertex path Pn for n=1,2,…,7. One can readily check that χgvt(P1)=1 and χgvt(Pn)=2 for n=2,3 and χgvt(Pn)=3 for n=4,5,6,7.
Theorem 4.
Let G be a graph without isolated vertices and isolated edges. Then
(5)χgvt(G)≤χgvd′(G).
Proof.
Suppose that f is a k-GVDEC of G. For any u∈V(G), let f(u)=f(uv), where uv∈E(G). Obviously, f is a k-GVDTC of G.
We now turn to investigating the general vertex-distinguishing total chromatic number of an n-vertex path.
Theorem 5.
Let Pn be a path on n vertices, n≥1. Then
(6)χgvt(Pn)=⌈3n+9n2+125273+3n-9n2+125273⌉.
Proof.
Denote by Pn=v1v2⋯vn a path Pn with vertex set {v1,v2,…,vn} and edge set {v1v2,v2v3,…,vn-1vn}. Let χgvt(Pn)=k, and let f be a k-GVDTC of Pn. Let αk=(k-11)+(k-12)+(k-13), βk=(k1)+(k2)+(k3), γk=1+(k-11)+(k-12), and
(7)k*=⌈3n+9n2+125273+3n-9n2+125273⌉.
Evidently, |Cf(vi)|≤3, and n≤βk (which implies k≥k*). In order to prove the conclusion, k=k*, it suffices to give a k*-GVDTC of Pn. When n≤7, it is not hard to construct the corresponding general vertex-distinguishing total colorings. Let n≥8. We first construct a k*-GVDTCf′ of Pβk recursively. Note that when n=βk, it has k*=k.
Procedure 1. Construct a 4-GVDTCf4 of Pβ4 (i.e., P14) as follows: the vertices v1,v2,…,v14 are colored by 1,3,4,2,3,1,4,2,3,4,3,4,4,1, respectively; the edges v1v2,v2v3,v3v4,…,v13v14 are colored by 1,3,1,1,2,4,2,2,2,3,3,4,4, respectively. It is easy to see that f4 is a 4-GVDTC of P14.
Procedure 2. Construct a k-GVDTCfk of Pβk based on a (k-1)-GVDTCfk-1 of Pβk-1. Let fk be
(8)fk(vi+γk)=fk-1(vi)+1,i=1,2,…,αk-1;fk(vαk+γk)=fk-1(vαk)=1;fk(vi+γkvi+1+γk)=fk-1(vivi+1)+1,i=1,2,…,αk-1;fk(v1)=1,fk(vγk)=k;
v2,v3,…,vγk-1 are colored by
(9)k-1,k︸2elements,k-2,k-1,k︸3elements,…,2,3,…,k︸k-1elements;
when k is even, v1v2,v2v3,v3v4,…,vγkvγk+1 are colored by(10)1︸1element,k-1,1︸2elements,1,k-2,1︸3elements,k-3,1,k-3,1︸4elements,1,k-4,1,k-4,1︸5elements,⋮,1,k-(k-4),…,1,k-(k-4),1︸k-3elements,k-(k-3),1,…,k-(k-3),1︸k-2elements,2,1,…,2,1︸k-2elements,2,2︸2elements;
when k is odd, v1v2,v2v3,v3v4,…,vγkvγk+1 are colored by(11)1︸1element,k-1,1︸2elements,1,k-2,1︸3elements,k-3,1,k-3,1︸4elements,1,k-4,1,k-4,1︸5elements,⋮,k-(k-4),1,…,k-(k-4),1︸k-3elements,1,k-(k-3),…,1,k-(k-3),1︸k-2elements,1,2,…,1,2,1︸k-2elements,2,2︸2elements.It should be pointed out that when j is odd for j∈{1,2,…,k-3}, the colors’ form is k-j,1,k-j,1,…,k-j,1 with totally j+1 elements, and when j is even for j∈{1,2,…,k-3}, the colors’ form is 1,k-j,1,k-j,…,1,k-j,1 with j+1 elements in total.
According to fk-1, we can see that, for any i,j=γk+1,γk+2,…,βk-1, and i≠j, it follows that {2,k} is not a total color set of vertices vi, 1∉Cfk(vi), and Cfk(vi)≠Cfk(vj). In addition, C1,C2,…,Cγ-1 are as follows: {1} (1 item), {{1,k-1},{1,k-1,k}} (2 items), {{1,k-2},{1,k-2,k-1},{1,k-2,k}} (3 items),…,{{1,k-j},{1,k-j,k-j+1}, {1,k-j,k-j+2},…,{1,k-j,k}} (j+1 items),…, and {{1,2},{1,2,3},{1,2,4},…,{1,2,k}} (k-1 items). And Cfk(vγk)={2,k}. So, fk is a k-GVDTC of Pβk. We now show that Pn also has a k-GVDTC based on a k-GVDTC of Pβk, for (k-11)+(k-12)+(k-13)<n<βk.
Let r=β-n, and let f be a k-GVDTC of Pβk constructed by Procedures 1 and 2. We first delete r vertices v1,v2,…,vr from Pβk. Obviously, the resulting graph, denoted by vr+1vr+2⋯vβk, is isomorphic to Pn. Let f′ be f′(vivi+1)=f(vivi+1) for i=r+1,r+2,…,βk-1; f′(vi)=f(vi) for i=r+2,…,β; and f′(vr+1)=1. Then f′ is a k-GVDTC of Pn.
All the above show that the conclusion holds.
According to Theorem 5, we have the same conclusion on cycles. Let Cn=v1v2⋯vnv1 be an n-vertex cycle with vertex set {v1,v2,…,vn} and edge set {v1v2,v2v3,…,vn-1vn,vnv1}.
Corollary 6.
For any cycle Cn(n≥3), one has
(12)χgvt(Cn)=⌈3n+9n2+125273+3n-9n2+125273⌉.
Proof.
Let Cn=v1v2⋯vnv1 and Pn=Cn∖v1vn; let also
(13)k*=⌈3n+9n2+125273+3n-9n2+125273⌉
and let f be a k*-GVETC of Pn, constructed by the method of Theorem 5. Then we can extend f to a k*-GVETC of Cn by assigning color 1 to edge v1vn. So, the conclusion holds.
In the following (Theorem 7 to Theorem 9), we discuss the general vertex-distinguishing total chromatic number of some kinds of special trees. A star Sn is the complete bipartite graph K1,n (n≥1). A double star Sm,n is a tree containing exactly two vertices that are not leaves (which are necessarily adjacent). A tristar Sp,q,r is a tree with vertex set V(Sp,q,r)={ui∣i=0,1,…,p}∪{vi∣i=0,1,…,q}∪{wi∣i=0,1,…,l} and edge set E(Sp,q,r)={u0ui∣i=1,2,…,p}∪{v0vi∣i=1,2,…,q}∪{w0wi∣i=1,2,…,r}∪{u0v0,v0w0}, where p,q,r are positive integers.
Theorem 7.
For a star Sn(n≥1), one has
(14)χgvt(Sn)={2,n=1,2;⌈8n+1-12⌉,n≥3.
Proof.
When n=1,2, the conclusion is trivial. For n≥3, let χgvt(Sn)=k, and ⌈(8n+1-1)/2⌉=k′. Since, for any u∈V(Sn)∖u0 (u0 is the vertex with dSn(u0)≥2), |Cf(u)|≤2 for any k-GVDTCf of Sn, it follows that n≤(k1)+(k2); that is, k≥k′. In order to prove k=k′, we need to show that there exists a k′-GVDTC of Sn. Otherwise, let Sn* be the graph with minimum n* such that Sn* does not have a k′-GVDTC, where n*≤n. Let u be a vertex of degree 1 in Sn*. Consider the graph G′=Sn*-u, obtained from Sn* by deleting the vertex u and its incident edge. By the assumption of Sn*, G′ has a k′-GVDTC, denoted by f′. In addition, by interchanging the colors of some vertex and its incident edge appropriately, we can assume |Cf′(u0)|≥2. Since n*-1<(k′1)+(k′2), there is at least one set {a,b}, for a,b∈{1,2,…,k′}, which is not the total color set of the vertices of G′. So, on the basis of f′, in Sn* we can color u and its incident edge uu0 by a and b, respectively. Obviously, the resulting coloring is a k′-GVDTC of Sn*.
For two vertices u,v of a graph G, to identify these two vertices is to replace them by a single vertex (denoted by u-v in this paper) incident to all the edges which were incident in G to either u or v. The resulting graph is denoted by G/{u,v}. In what follows, we denoted by [1,k] the set of {1,2,…,k}.
Theorem 8.
Let Sm,n(m≥n≥1) be a double star, and l=m+n. Then
(15)χgvt(Sm,n)={3,l=2,3,4,5;4,l=6;⌈8l+1-12⌉,l≥7.
Proof.
When l≤6, the results are easy to be proved. When l≥7, let u,v be two vertices with degree more than 1, and G′=Sm,n/{u,v}. Evidently, the graph G′ is isomorphic to the star Sl. Let k′=⌈(8l+1-1)/2⌉. Since |Cf(x)|≤2 for any x∈V(Sm,n)∖{u,v}, we have χgvt(Sm,n)≥k′.
By Theorem 7, G′ contains a k′-GVDTCf′. Evidently, |Cf′(x)|≤2 for any x∈V(G′)∖{u-v} and |Cf′(u-v)|≤k′. If |Cf′(u-v)|≤2, then we can extend f′ to a k′-GVDTC of Sm,n by coloring vertices u,v and edge uv with any three different colors in [1,k]∖{f′(u-v)}; if |Cf′(u-v)|=l′≥3, we without loss of generality assume Cf′(u-v)∖f′(u-v)=[1,l′]. Let Eu (resp., Ev) be the set of edges (except edge uv) incident to u (resp., v) in Sm,n. We now extend f′ to a k′-GVDTC of Sm,n as follows. By the fact that there remain vertices u,v and edge uv uncolored in Sm,n when f′ is restricted to Sm,n, we consider the following two cases. First, one of u,v, say u, satisfies that Cf′(Eu) contains at most two elements. Assume Cf′(Eu)⊆{1,2}; we then color u,uv,v by 2,3,c, respectively, where c=4 when Cf′(Eu)≠{3,4} and c=2 when Cf′(Eu)={3,4}. The resulting coloring of Sm,n is also denoted by f′. Then it follows in Sm,n that |Cf′(u)|≥3, |Cf′(v)|≥3, and 4∉Cf′(u) and 4∈Cf′(v). So, f′ is a k′-GVDTC of Sm,n. Second, |Cf′(Ev)|≥3 and |Cf′(Eu)|≥3; then we will further discuss two subcases.
Consider |Cf′(Eu)|=|Cf′(Ev)|=k′. Suppose that Vu (and Vv) is the set of vertices, except v (or u), adjacent to u (and v) in Sm,n. Because f′ is a k′-GVDTC of G′, either Vu or Vv contains no vertices with total color set {i}, for some i∈{1,2,…,k′}. Without loss of generality we assume that there is no vertex x∈Vu with Cf′(x)={i}. For any vertex y in Vu such that Cf′(y)={i,j} and f′(yu)=i, interchange the two colors of y and yu. The resulting coloring, still denoted by f′, satisfies that Cf′(Eu) does not contain color i. Then we color u,uv,v by any three colors in {1,2,…,k′}∖{i} and obtain a k′-GVDTC of Sm,n.
Consider |Cf′(Eu)|<k′ or |Cf′(Ev)|<k′; assume |Cf′(Eu)|<k′ here. Let i∉Cf′(Eu). Color v by i and color u,uv by any two colors in [1,k′]∖{i}. Obviously, the resulting coloring is a k′-GVDTC of Sm,n.
All the above show that χgvt(Sm,n)≤k′. So, the conclusion holds.
Theorem 9.
Let Sp,q,l be a tristar defined as above, and l=p+q+r. Then
(16)χgvt(Sp,q,r)={3,l=3,4;4,l=5,6;⌈(8r+1-1)/2⌉,l≥7.
Proof.
When l=3,4, the conclusion is easy to be checked; when l=5 or 6, since |V(Sp,q,r)|≥8 and (31)+(32)+(33)=7, it follows that χgvt(Sp,q,r)>3. In addition, it is not hard to give a 4-GVDTC of V(Sp,q,r) in each case of l=5 or 6, so χgvt(Sp,q,r)=4; when l≥7, let k′=⌈(8r+1-1)/2⌉. Identify vertices u0 and v0 in Sp,q,r and let G′=Sp,q,r/{u,v}. By Theorem 8, G′ has a k′-GVDTCf′. With the analogous analysis method of Theorem 8, we can also extend f′ to a k′-GVDTC of Sp,q,r. This shows χgvt(Sp,q,r)≤k′. On the other hand, for any k-GVDTCf of Sp,q,r, it has that l≤(k1)+(k2); that is, k≥k′. So, the result holds.
In the above, we construct a k-GVDTC of a graph G by extending a k-GVDTC of graph G′, where G′ is the resulting graph of identifying two vertices of degree more than 1 in G. But this method does not always work. For instance, the graph G/{u,v} shown in Figure 1(b) has a 4-GVDTC, but the graph G shown in Figure 1(a) does not contain any 4-GVDTC. So any 4-GVDTC of G/{u,v} can not be extended to a 4-GVDTC of G.
(a) A graph, G; (b) G/{u,v}.
In the following we are devoted to the study of the general vertex-distinguishing chromatic number of fan graph Fn, wheel graph Wn, and complete graph Kn. Let G,H be two graphs such that V(G)∩V(H)=∅. The join G+H of G and H is a graph with vertex set V(G+H)=V(G)∪V(H) and edge set E(G+H)=E(G)∪E(H)∪{uv∣u∈V(G),v∈V(H)}. A fan graph Fn is defined as the join of a path of n vertices and an isolated vertex. A wheel graph Wn is defined as the join of a cycle of n vertices and an isolated vertex.
Theorem 10.
Let Fn be a fan, n≥2; then
(17)χgvt(Fn)={3,n=2,3,…,6;4,n=7,8,…,14;k,n≥15,
where (k-11)+(k-12)+(k-13)+(k-14)<n≤(k1)+(k2)+(k3)+(k4).
Proof.
Let V(Fn)={vi∣i=0,1,…,n} and E(Fn)={v1v2,v2v3,…,vn-1vn}∪{v0vi∣i=1,2,…,n}. When n≤14, the conclusion is easy to show. We now consider the case of n≥15 (which implies k≥5). Since (k-11)+(k-12)+(k-13)+(k-14)<n and |Cf(vi)|≤4 for i∈[1,n], we can easily deduce that χgvt(Fn)≥k. So, it suffices to show that Fn contains a k-GVDTC. In particular, we prove that Fn contains a k-GVDTC such that the total color set of v0 contains at least 5 elements. By induction on n. When n=15, it is not hard to construct such a k(≥5)-GVDTC of Fn. Suppose that, for any Fn′, n′<n, there exists a k-GVDTC of Fn′. Consider the fan graph Fn-1, and let f be a k-GVDTC of Fn-1. Anyway, we can assume that, for any color x∈[1,k], there is an edge vivi+1 for some i∈[1,n-1] such that f(vivi+1)=x (If not, we can permutate x and f(vivi+1)).
Note that for any edge vivi+1 of Fn-1, i∈[1,k], if we replace this edge by a vertex u and connect this vertex to v0, vi, and vi+1, then the resulting graph, denoted by Fn-1vi,u,vi+1, is isomorphic to Fn. We will use this to construct a k-GVDTC of Fn based on f. It is obvious that there remain only 4 uncolored elements, viu,uvi+1,uv0, and u in Fn-1vi,u,vi+1, if we restrict f to Fn-1vi,u,vi+1. We need to consider the following 2 cases.
Case 1. There exist colors x,y,z∈[1,k] such that {x,y,z}∉Cf(Fn-1). Let vivi+1 be the edge with f(vivi+1)=x, i∈[1,n-1]. In Fn-1vi,u,vi+1, let f(uvi)=f(uvi+1)=x, f(uv0)=y, and f(u)=z, and the resulting coloring is still denoted by f. Evidently, in Fn-1vi,u,vi+1, Cf(u)={x,y,z}; meanwhile Cf(vi) and Cf(vi+1) are the same as those in Fn-1, and |Cf(v0)|≥5. So, f a k-GVDTC of Fn-1vi,u,vi+1.
Case 2. Four different colors x,y,z,w∈[1,k] such that {x,y,z,w}∉Cf(Fn-1). Select an edge vivi+1, i∈[1,n-1], for which f(vivi+1)=x. Since f is a k-GVDTC of Fn-1, Cf(vi)∖Cf(vi+1) contains at least one element (here we assume |Cf(vi)|≥|Cf(vi+1)|), say c. Obviously, c≠x, c∈Cf(vi), and c∉Cf(vi+1) in Fn-1. We can permutate the colors so that c∈{y,x,w} and f(vi)=c in Fn-1, say c=y. Then, in Fn-1vi,u,vi+1, erase the color of vertex vi and recolor it by color x, and let f(uvi)=y, f(uvi+1)=x, f(uv0)=z, and f(u)=w. Obviously, in Fn-1vi,u,vi+1, it follows that Cf(u)={x,y,z,w}, Cf(vi) and Cf(vi+1) are the same as those in Fn-1, and |Cf(v0)|≥5. So, f a k-GVDTC of Fn-1vi,u,vi+1.
All of the above show that Fn has a k-GVDTC.
Theorem 11.
Let Wn be a wheel graph, n≥2; then
(18)χgvt(Wn)={3,n=2,3,…,6;4,n=7,8,…,14;k,n≥15,
where (k-11)+(k-12)+(k-13)+(k-14)<n≤(k1)+(k2)+(k3)+(k4).
We omit the proof for Theorem 11, since it is analogous to that of Theorem 10.
Theorem 12.
For a complete graph Kn, n≥1, one has
(19)χgvt(Kn)=1+⌈log2n⌉.
Proof.
When n<10 the conclusion is easy to show. So we assume n≥10.
Denote by V(Kn)={v1,v2,…,vn} and {1,2,…,k} the set of k colors. For integer l=⌈n/2⌉, we construct a l-GVDTCf′ of Kn as follows: let f′(vi)=i for any i∈[1,l]; f′(vivj)=1 for i≠j, i∈[1,l], and j∈[1,n]; f(vi)=i-l+2 for i∈[l+1,n-2]; f′(vivj)=2 for i≠j, i∈[l+1,n-2], and j∈[l+1,n]; f′(vn-1vn)=3; f′(vn-1)=4; f′(vn)=5. One can readily check that Cf′(v1)={1}, Cf′(vi)={1,i} for i=1,2,…,l; Cf′(vi)={1,2,i-l+2} for i=l+1,l+2,…,n-2; Cf′(vn-1)={1,2,3,4}; and Cf′(vn-1)={1,2,3,5}. Thus, f′ is a l-GVDTC of Kn, which shows χgvt(Kn)≤l.
Suppose that χgvt(Kn)=k and f is a k-GVDTC of Kn. Since for any two vertices vi,vj (i≠j∈[1,k]), Cf(vi)∩Cf(vj)≠∅, one can see that there is at most one vertex whose total color set contains only one color. If there is a vertex v with |C(v)|=1, without loss of generality assume C(v)={1}; then the total color set of each vertex contains color 1, which indicates
(20)n≤1+(k-11)+(k-12)+⋯+(k-1k-1)=2k-1.
If there is no vertex whose total color set contains only one color, then for each vi it has |C(vi)|≥2. Since, for any vertex vj≠vi, |C(vi)∩C(vj)|≥1, it follows that C(v) and [1,k]∖C(v) can not be two total color sets with respect to f. This implies n≤2k-1. So k≥1+⌈log2n⌉.
To prove k=1+⌈log2n⌉, we need to show that Kn has a (1+⌈log2n⌉)-GVDTC. In particular, we show that any Kn has a (1+⌈log2n⌉)-GVDTC such that for each color c∈[1,1+⌈log2n⌉] there is at least one vertex in Kn being colored by c.
We prove this by induction on n. When n=10, 5-GVDTC is the f′ defined above for n=10,l=5, where Cf′(v1)={1}. Consider the graph Kn-vn obtained from Kn by deleting vertex vn and its incident edges. Obviously, Kn-vn is isomorphic to Kn-1. By the induction hypothesis, Kn-1 has a (1+⌈log2n-1⌉)-GVDTC, say f, such that for each color c∈[1,1+⌈log2n-1⌉] there is at least one vertex of Kn-1 being colored by c. Since n≤2k-1, there must be some set C (denoted by {c1,c2,…,cl})∉Cf(Kn-1). We consider the following two cases.
Consider ⌈log2n-1⌉=⌈log2n⌉. By the induction hypothesis, each color ci∈[1,1+⌈log2n-1⌉] appears at a vertex. Without loss of generality assume f(vi)=ci for i=1,2,…,l. Then, f is extended to a (1+⌈log2n-1⌉)-GVDTC of Kn via coloring vn by one of the colors in [c1,c2,…,cl]; coloring vnvi for i=1,2,…,l by ci; and coloring vnvj for j=l+1,…,n-1 by one of the colors in Cf(vj) (vj∈V(Kn-1)).
Consider ⌈log2n-1⌉=⌈log2n⌉-1. Then on the basis of f, we only need to color vn and all of its incident edges in Kn by color 1+⌈log2n-1⌉.
One can readily check that the resulting coloring of Kn in the above two cases is (1+⌈log2n⌉)-GVDTCs of Kn such that each color in [1,1+⌈log2n⌉] appears at a vertex of Kn. Hence, Kn has a (1+⌈log2n⌉)-GVDTC, and the conclusion holds.
In the following, we present a trivial upper bound of the general vertex-distinguishing total chromatic number of the join graph of two graphs.
Theorem 13.
Suppose G, H are two simple graphs and G∩H=∅. Then
(21)χgvt(G+H)≤χgvt(G)+χgvt(H).
Proof.
Let V(G)={ui∣i=1,2,…,m} and V(H)={vi∣i=1,2,…,n}. Suppose that f1 is a χgvt(G)-GVDTC of G and f2 is a χgvt(H)-GVDTC of H, where the sets of colors of f1 and f2 are C1 and C2 (C1∩C2=∅), respectively.
Define f as f(uivj)=f1(ui)(orf2(vj)),i=1,2,…,m;j=1,2,…,n.
Combining colorings f1,f2,f, we can obtain a (χgvt(G)+χgvt(H))-GVDTC of G+H.
3. Remarks
Based on the above results, we propose two conjectures as follows.
Conjecture 14.
Let G be a graph without isolated vertices. Then
(22)χgvt(G)≤⌈n2⌉.
Conjecture 15.
Let G be a connected graph on n vertices. Then
(23)χgvt(G)≤1+⌈log2n⌉.
Note that if Conjecture 15 is true, then Conjecture 14 is true. On the other hand, if Conjecture 14 is true, then the upper bound cannot be improved. For instance, the graph G contains exactly three K2 components. It is easy to show that χgvt(G)=3.
In addition, there is a very interesting observation about the general vertex-distinguishing total chromatic number.
Observation 1.
Let H be a subgraph of a graph G. Then it possibly follows that
(24)χgvt(H)>χgvt(G).
As an illustration of this observation, we consider the path P15 and the fan graph F14. P15 is a subgraph of F14, while χgvt(P15)(=5)>χgvt(F14)(=4).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant nos. 61127005 and 61309015) and the National Basic Research Program of China (973 Program) (no. 2010CB328103).
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