Edge Double-Critical Graphs

Problem statement: The vertex double-critical conjecture that the onl y vertex doublecritical graph is the complete graph has remained u nresolved for over forty years. The edge analogue of this conjecture has been proved. Approach: It was observed that if the chromatic number decre ases by two upon the removal of a 2-matching, then the 2 -matching comprises four vertices which determine an induced subgraph isomorphic to the com plete graph on four vertices. This observation was generalized to t-matchings. Results: In this note, it has been shown that the only edge oublecritical graph is the complete graph. Conclusion/Recommendations: An alternate proof that the only edge double-critical graph is the complete graph ha s been obtained. Moreover, the result has been obtained independently.


INTRODUCTION
The graphs considered in this study are finite, undirected and simple. For a given graph G, the vertex and edge sets of G are denoted by V(G) and E(G), respectively. The order of G, denoted by n = |V(G)|, is the cardinality of V(G). An r-clique is a complete subgraph of order r and is denoted by K r . A subset M of E(G) is said to be independent whenever no two edges in M share a common vertex. In case |M| = k, the set M is called a k-matching. For a subset X of V(G), the subgraph of G induced by X is denoted by G[X]. All vertex colorings are proper, i.e., a partition of V(G) into independent subsets of V(G) called color classes. Lastly, χ(G) denotes the chromatic number of G and is the minimum cardinality of a partition of V(G) determined by a proper vertex coloring of G.
A graph G is said to be vertex double-critical provided χ(G-v) = χ(G)-2 for every adjacent pair of vertices u, v. This definition arises out of its relation to the Erdos-Lovasz Tihany Conjecture. A special case of this conjecture is that the only vertex double-critical graph is the complete graph; it is often referred to as the Erdos-Lovasz double-critical conjecture. A brief discussion of the Erdos-Lovasz Tihany Conjecture and related results for quasi-line graphs are given in (Balogh et al., 2009). Stiebitz (1987) has shown that K 5 is the only 5-chromatic vertex double-critical graph. To date, the Erdos-Lovasz double-critical conjecture remains open for k-chromatic graphs with k≥6. In Theorem 6 of (Kawarabayashi et al., 2010), the edge analogue of the Erdos-Lovasz double-critical conjecture is proved. This note offers an independent proof of the edge analogue of the Erdos-Lovasz double-critical conjecture. Relations to FTTMs and the inertia tensor of a tetrahedron as defined in (Ahmad et al., 2010;Tonon, 2005), respectively are also being investigated.

Edge double-critical graphs:
It is now shown that K n is the only edge double-critical graph. First, some notational conventions and a required definition are given. Let M t = {e 1 , e 2 , …, e t } be a set of t edges in E(G) and set e i = u i v i for i = 1, 2, …, t. Next, define M t Clearly, M t is a tmatching when |M t | = 2t.
Necessarily, an edge double-critical graph is connected. An important observation is given in the following lemma. Proof: Set k = χ(G) and let e i = u i v i for i = 1, 2. Consider any (k-2)-coloring of G-M 2 , the colors being from among {c 1 , c 2 , …, c k -2 }. Then u 1 and v 1 must be colored the same since otherwise there would exist a (k-2)-coloring of G-e 2 . A similar argument shows that u 2 and v 2 must be colored the same, necessarily using a different color from that used for u 1 and v 1 . Next, observe that u 1 u 2 ∈ E(G-M 2 ). Else, both u 1 and u 2 could be recolored using color c k -1 . But this would allow e 1 and e 2 to be added back to G-M 2 resulting in a coloring of G using fewer than k colors. A similar argument shows that u 1 v 2 , v 1 u 2 , v 1 v 2 ∈ E(G-M 2 ). Consequently, G[M 2 * ] ≅ K 4 .
Proof: Let k = χ(G). The result is trivial for t = 1. Let t ≥ 2 and consider a subset M t of E(G) such that χ(G-M t ) = k-t. Because |M t | = t, it follows that M t is a t-matching as incident edges can decrease the chromatic number of a graph by at most one upon their removal. Observe now that for all pairs i, j with i ≠ j, χ(G-e i -e j ) = k-2. By setting M 2 (i, j) = {e i , e j } and applying Lemma 1, Proposition 1: Every t-matching in K 2t is critical.
Corollary 1: Every matching in K n , n ≥ 2, is critical. Lemma 1 and Corollary 1 together set the stage for the main result of this note.
Theorem 2: G is edge double-critical if and only if G ≅ K n , provided n ≥ 4.
Proof: If G ≅ K n , where n ≥ 4, then by Corollary 1, every 2-matching in K n is critical. Thus, G is edge double-critical. Conversely, let G by a connected, edge double-critical graph. Take any u, v∈V(G) and suppose to the contrary that uv ∉ E(G). Then N(u) = N(v) = {w u, v } for some vertex w u,v ∈ V(G). Otherwise, because G is connected, it would follow that u, v ∈ M 2 * for some 2-matching M 2 . Since G is edge doublecritical, G[M 2 * ] ≅ K 4 by Lemma 1. This implies that uv ∈ E(G), contrary to our supposition. Next, observe that N(z) = {w u, v } for every vertex z ≠ w u, v . Else, by using exactly the same argument as above, we would be forced to conclude that z ∈ N(u) = {w u, v }, which is clearly not possible by the choice of z. The above argument leads to the conclusion that G is a star. But such a graph is known not to be edge double-critical because of the absence of 2-matchings in any star. Hence, uv ∈ E(G) so that G ≅ K n .