Multi-bridge graphs are anti-magic

An anti-magic graph is a graph whose | E | edges can be labeled with the ﬁrst | E | natural numbers such that each edge receives a distinct number and each vertex receives a distinct vertex sum which is obtained by taking the sum of the labels of all the edges incident to it. We prove that the multi-bridge graph is anti-magic.


Introduction
Let G = (V, E) be a graph with neither loop nor multiple edges. An anti-magic labeling of G is a bijection ϕ from E to {1, 2, . . . , |E|} such that the sum of the labels on the edges incident to a vertex, called the vertex sum, is distinct for each vertex. A graph is anti-magic if it admits an anti-magic labeling.
The concept of anti-magic graphs has its origin from the book [7] where Hartsfield and Ringel conjectured that all connected graphs but the single edge K 2 are anti-magic. Since then, the problem of deciding which graphs are anti-magic has attracted much attention. Nevertheless the conjecture remains unsettled despite concerted efforts by mathematicians in graph theory.
In the same book, Hartsfield and Ringel remarked that even when the conjecture is restricted to trees, no complete affirmative answer has been offered. Some results concerning the antimagicness of trees are given in [8] and [9].
On the other hand, by confining the attention on regular graphs, the situation turns out to be a lot more delightful. In [4], Cranston showed that every regular bipartite graph with degree at least 2 is anti-magic. In [5], Cranston et al. proved that Hartsfield and Ringel's conjecture is true for all odd regular graphs. Shortly afterwards, in [3], Chang et al. proved that all even regular graphs are anti-magic. By modifying the argument used in [5], Bérczi et al. in [2] also proved that even regular graphs are anti-magic. For more details on anti-magic graphs, we refer the reader to [6]. For some recent results on anti-magic graphs, we refer the reader to [10].
In view of this, we turn our attention to graphs which are close to being regular. Consider a graph with only two vertices and having r multiple edges joining them, r ≥ 3. Subdivide the edges of this graph arbitrarily so that at most one edge is not subdivided. Call the result graph an r-bridge graph and denote it by θ(m 1 , m 2 , . . . , m r ) if the lengths of the paths are m 1 , m 2 , . . . , m r respectively.
The purpose of this paper is to prove the following result.
Theorem 1.1. Every r-bridge graph is anti-magic.
In a forth-coming paper, we shall make use of the above result to prove the anti-magicness of a class of not quite regular graphs. Hence it is an appetizer result for a more general result which is to appear later.
We note in passing that in [1], Alon et al. proved that all dense graphs are anti-magic while in [11], Wang initiated the investigation on the anti-magicness of sparse graphs. Incidentally, the graphs in this paper and those in our forth-coming papers are sparse graphs.
Let x and y denote the two vertices of degree r in θ(m 1 , m 2 , . . . , m r ) and let w(x), w(y) denote the vertex sums of x, y respectively.
The proof is divided into three cases.
The labelings depicted in Figure 1 show that if m 1 ≤ 2, the 3-bridge graph is anti-magic. Hence we assume that m 1 ≥ 3.    Note that the vertex sums of the degree-2 vertices consist of distinct odd natural numbers and that the vertex sums of x and y are both even and are given by w(x) = 2(m 1 + m 3 + 1) and w(y) = 2m 1 + m 1 + m 2 + m 3 + 1 respectively.
This shows that ϕ 0 is an anti-magic labeling of the 3-bridge graph. In this case, an anti-magic labeling is obtained by swapping the labels m 1 − 1, m 1 (on the last two edges of the m 1 -path) from the anti-magic labeling ϕ 0 given in Subcase 1.1. Note that there are only three vertices whose vertex-sums are even, namely x, y and the second last vertex on the m 1 -path. Since the vertex-sums are 2(m 1 +m 3 +1), 2m 1 +m 1 +m 2 +m 3 and 2m 1 −2 respectively, they are distinct natural numbers.
This completes the proof for Case 1.  Note that the vertex sums w(x) and w(y) of x and y are given by 3m 1 + 2m 2 + 2m 3 + m 4 + 2 and 4m 1 + 3m 2 + m 3 + 2 respectively. Note that the vertex sums of the degree-2 vertices consist of distinct natural odd numbers and they are all less than either of w(x) and w(y).
This means that ϕ 1 is an anti-magic labeling of the 4-bridge. In this case, an anti-magic labeling is obtained by labeling the edges of the i-th path with the labels (i − 1)m + 1, (i − 1)m + 2, . . . , im successively all starting from x to y. In this case w(x) = 6m + 4 and w(y) = 10m. The rest of the vertex sums consist of distinct odd natural numbers. Figure 4(ii) illustrates the case m = 3.
(iii) Finally, label the edges of the m 3i -path with p i−1 + m 3i−1 + m 3i+1 + 1, p i−1 + m 3i−1 + m 3i+1 + 2, . . . , p i−1 + m 3i−1 + m 3i+1 + m 3i starting from the vertex x. It is routine to check that the vertex sums of x and y are given by Also, note that the vertex sums of the degree-2 vertices consist of distinct odd natural numbers each of which is less than either of w(x) and w(y).
Clearly the vertex sums of the degree-2 vertices in ϕ 2 consist of odd distinct natural numbers and each is less than either of w(x) and w(y).
Hence ϕ 2 is an anti-magic labeling of the 5-bridge. Now suppose k ≥ 2.
Also, note that the vertex sums of the degree-2 vertices consist of distinct odd natural numbers each of which is less than either of w(x) and w(y).
This completes the proof for Case 3 and so is the proof for Theorem 1.1.